Logistic distribution

Percentage Accurate: 99.6% → 99.7%

Time: 24.7s

Alternatives: 12

Speedup: 1.5×

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: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\left(2 + e^{\frac{\left|x\right|}{s}}\right) + e^{\frac{-\left|x\right|}{s}}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-*l* 99.6%

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

   4. times-frac 99.6%

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

   5. associate-*r/ 99.6%

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

   6. associate-/l* 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. exp-neg 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around 0 99.7%

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

   1. +-commutative 99.7%

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

   2. mul-1-neg 99.7%

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

   3. distribute-frac-neg 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 97.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{x}{s} \cdot \frac{x \cdot -0.25}{s}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (exp (fma -2.0 (log 2.0) (* (/ x s) (/ (* x -0.25) s)))) s))

C

float code(float x, float s) {
	return expf(fmaf(-2.0f, logf(2.0f), ((x / s) * ((x * -0.25f) / s)))) / s;
}

Julia

function code(x, s)
	return Float32(exp(fma(Float32(-2.0), log(Float32(2.0)), Float32(Float32(x / s) * Float32(Float32(x * Float32(-0.25)) / s)))) / s)
end

Derivation

1. Initial program 99.6%

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

   1. associate-*l* 99.6%

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

   2. +-commutative 99.6%

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

   3. +-commutative 99.6%

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

3. Simplified 99.6%

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

   1. add-exp-log 98.0%

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

   2. log-div 98.0%

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

   3. add-log-exp 98.0%

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

   4. *-commutative 98.0%

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

   5. log-prod 97.8%

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

5. Applied egg-rr 97.8%

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

   1. associate--r+ 97.8%

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

   2. exp-diff 98.0%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around inf 56.5%

   \[\leadsto \frac{e^{\color{blue}{-2 \cdot \log 2 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}}}{s} \]
9. Step-by-step derivation

   1. +-commutative 56.5%

      \[\leadsto \frac{e^{\color{blue}{\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}}{s} \]

   2. mul-1-neg 56.5%

      \[\leadsto \frac{e^{\left(\color{blue}{\left(-\frac{\left|x\right|}{s}\right)} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}{s} \]

   3. distribute-frac-neg 56.5%

      \[\leadsto \frac{e^{\left(\color{blue}{\frac{-\left|x\right|}{s}} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}{s} \]

10. Simplified 90.5%

    \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(-2, \log 2, \frac{-0.25 \cdot \left(x \cdot x\right)}{s \cdot s}\right)}}}{s} \]

11. Taylor expanded in x around 0 90.5%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \color{blue}{-0.25 \cdot \frac{{x}^{2}}{{s}^{2}}}\right)}}{s} \]
12. Step-by-step derivation

    1. unpow2 90.5%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, -0.25 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}\right)}}{s} \]

    2. associate-*r/ 90.5%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \color{blue}{\frac{-0.25 \cdot \left(x \cdot x\right)}{{s}^{2}}}\right)}}{s} \]

    3. *-commutative 90.5%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{\color{blue}{\left(x \cdot x\right) \cdot -0.25}}{{s}^{2}}\right)}}{s} \]

    4. associate-*l* 90.5%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{\color{blue}{x \cdot \left(x \cdot -0.25\right)}}{{s}^{2}}\right)}}{s} \]

    5. unpow2 90.5%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{x \cdot \left(x \cdot -0.25\right)}{\color{blue}{s \cdot s}}\right)}}{s} \]

    6. times-frac 95.9%

       \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \color{blue}{\frac{x}{s} \cdot \frac{x \cdot -0.25}{s}}\right)}}{s} \]

13. Simplified 95.9%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \color{blue}{\frac{x}{s} \cdot \frac{x \cdot -0.25}{s}}\right)}}{s} \]

14. Final simplification 95.9%

    \[\leadsto \frac{e^{\mathsf{fma}\left(-2, \log 2, \frac{x}{s} \cdot \frac{x \cdot -0.25}{s}\right)}}{s} \]

Alternative 3: 95.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2 \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* 2.0 (fma s (exp (/ (fabs x) s)) s))))

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(2.0) * fma(s, exp(Float32(abs(x) / s)), s)))
end

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-/l* 99.6%

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

   4. distribute-frac-neg 99.6%

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

   5. exp-neg 99.6%

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

   6. associate-/r/ 99.6%

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

   7. /-rgt-identity 99.6%

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

   8. associate-*l* 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around inf 94.3%

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

5. Final simplification 94.3%

   \[\leadsto \frac{1}{2 \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]

Alternative 4: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (fabs x) 4.999999841327613e-21)
   (/ 1.0 (fma s 4.0 (* x (/ x s))))
   (/ (exp (/ (- (fabs x)) s)) s)))

C

float code(float x, float s) {
	float tmp;
	if (fabsf(x) <= 4.999999841327613e-21f) {
		tmp = 1.0f / fmaf(s, 4.0f, (x * (x / s)));
	} else {
		tmp = expf((-fabsf(x) / s)) / s;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (abs(x) <= Float32(4.999999841327613e-21))
		tmp = Float32(Float32(1.0) / fma(s, Float32(4.0), Float32(x * Float32(x / s))));
	else
		tmp = Float32(exp(Float32(Float32(-abs(x)) / s)) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (fabs.f32 x) < 4.99999984e-21

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.1%

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

      5. associate-*r/ 99.3%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.0%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. mul-1-neg 99.4%

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

      3. distribute-frac-neg 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in s around inf 76.7%

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

      1. associate-+r+ 76.7%

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

      2. distribute-lft1-in 76.7%

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

      3. metadata-eval 76.7%

         \[\leadsto \frac{1}{\color{blue}{0} \cdot \left|x\right| + \left(4 \cdot s + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      4. mul0-lft 76.7%

         \[\leadsto \frac{1}{\color{blue}{0} + \left(4 \cdot s + \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      5. unpow2 76.7%

         \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      6. sqr-abs 76.7%

         \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{\color{blue}{x \cdot x}}{s}\right)} \]

      7. unpow2 76.7%

         \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{\color{blue}{{x}^{2}}}{s}\right)} \]

      8. remove-double-neg 76.7%

         \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{\color{blue}{-\left(-{x}^{2}\right)}}{s}\right)} \]

      9. mul-1-neg 76.7%

         \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{-\color{blue}{-1 \cdot {x}^{2}}}{s}\right)} \]

      10. *-commutative 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{-\color{blue}{{x}^{2} \cdot -1}}{s}\right)} \]

      11. metadata-eval 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{-{x}^{2} \cdot \color{blue}{\left(-2 - -1\right)}}{s}\right)} \]

      12. distribute-rgt-out-- 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{-\color{blue}{\left(-2 \cdot {x}^{2} - -1 \cdot {x}^{2}\right)}}{s}\right)} \]

      13. neg-mul-1 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \frac{\color{blue}{-1 \cdot \left(-2 \cdot {x}^{2} - -1 \cdot {x}^{2}\right)}}{s}\right)} \]

      14. associate-*r/ 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \color{blue}{-1 \cdot \frac{-2 \cdot {x}^{2} - -1 \cdot {x}^{2}}{s}}\right)} \]

      15. associate-*r/ 76.7%

          \[\leadsto \frac{1}{0 + \left(4 \cdot s + \color{blue}{\frac{-1 \cdot \left(-2 \cdot {x}^{2} - -1 \cdot {x}^{2}\right)}{s}}\right)} \]

   9. Simplified 78.7%

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

   if 4.99999984e-21 < (fabs.f32 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. associate-*l* 99.7%

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

      2. +-commutative 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. add-exp-log 99.4%

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

      2. log-div 99.4%

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

      3. add-log-exp 99.4%

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

      4. *-commutative 99.4%

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

      5. log-prod 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. associate--r+ 99.4%

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

      2. exp-diff 99.5%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in s around 0 94.9%

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

      1. mul-1-neg 94.9%

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

      2. distribute-frac-neg 94.9%

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

   10. Simplified 94.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(s, 4, x \cdot \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \end{array} \]

Alternative 5: 94.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-2 \cdot \log 2 + \frac{-0.25}{\frac{s \cdot s}{x \cdot x}}}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 4.999999999099794e-24)
   (/ (exp (/ (- (fabs x)) s)) s)
   (/ (exp (+ (* -2.0 (log 2.0)) (/ -0.25 (/ (* s s) (* x x))))) s)))

C

float code(float x, float s) {
	float tmp;
	if (s <= 4.999999999099794e-24f) {
		tmp = expf((-fabsf(x) / s)) / s;
	} else {
		tmp = expf(((-2.0f * logf(2.0f)) + (-0.25f / ((s * s) / (x * x))))) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 4.999999999099794e-24) then
        tmp = exp((-abs(x) / s)) / s
    else
        tmp = exp((((-2.0e0) * log(2.0e0)) + ((-0.25e0) / ((s * s) / (x * x))))) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(4.999999999099794e-24))
		tmp = Float32(exp(Float32(Float32(-abs(x)) / s)) / s);
	else
		tmp = Float32(exp(Float32(Float32(Float32(-2.0) * log(Float32(2.0))) + Float32(Float32(-0.25) / Float32(Float32(s * s) / Float32(x * x))))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(4.999999999099794e-24))
		tmp = exp((-abs(x) / s)) / s;
	else
		tmp = exp(((single(-2.0) * log(single(2.0))) + (single(-0.25) / ((s * s) / (x * x))))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 5e-24

   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. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. add-exp-log 98.6%

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

      2. log-div 98.6%

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

      3. add-log-exp 98.6%

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

      4. *-commutative 98.6%

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

      5. log-prod 98.4%

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

   5. Applied egg-rr 98.4%

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

      1. associate--r+ 98.4%

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

      2. exp-diff 98.6%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in s around 0 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-frac-neg 93.0%

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

   10. Simplified 93.0%

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

   if 5e-24 < s

   1. Initial program 99.5%

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

      1. associate-*l* 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.6%

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

      1. add-exp-log 97.6%

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

      2. log-div 97.6%

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

      3. add-log-exp 97.6%

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

      4. *-commutative 97.6%

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

      5. log-prod 97.4%

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

   5. Applied egg-rr 97.4%

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

      1. associate--r+ 97.4%

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

      2. exp-diff 97.6%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in s around inf 67.9%

      \[\leadsto \frac{e^{\color{blue}{-2 \cdot \log 2 + \left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right)}}}{s} \]
   9. Step-by-step derivation

      1. +-commutative 67.9%

         \[\leadsto \frac{e^{\color{blue}{\left(-1 \cdot \frac{\left|x\right|}{s} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}}{s} \]

      2. mul-1-neg 67.9%

         \[\leadsto \frac{e^{\left(\color{blue}{\left(-\frac{\left|x\right|}{s}\right)} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}{s} \]

      3. distribute-frac-neg 67.9%

         \[\leadsto \frac{e^{\left(\color{blue}{\frac{-\left|x\right|}{s}} + \left(\frac{\left|x\right|}{s} + -1 \cdot \frac{0.5 \cdot {\left(\left|x\right|\right)}^{2} + -0.25 \cdot {\left(\left|x\right|\right)}^{2}}{{s}^{2}}\right)\right) + -2 \cdot \log 2}}{s} \]

   10. Simplified 96.5%

       \[\leadsto \frac{e^{\color{blue}{\mathsf{fma}\left(-2, \log 2, \frac{-0.25 \cdot \left(x \cdot x\right)}{s \cdot s}\right)}}}{s} \]
   11. Step-by-step derivation

       1. fma-udef 96.5%

          \[\leadsto \frac{e^{\color{blue}{-2 \cdot \log 2 + \frac{-0.25 \cdot \left(x \cdot x\right)}{s \cdot s}}}}{s} \]

       2. associate-/l* 96.5%

          \[\leadsto \frac{e^{-2 \cdot \log 2 + \color{blue}{\frac{-0.25}{\frac{s \cdot s}{x \cdot x}}}}}{s} \]

   12. Applied egg-rr 96.5%

       \[\leadsto \frac{e^{\color{blue}{-2 \cdot \log 2 + \frac{-0.25}{\frac{s \cdot s}{x \cdot x}}}}}{s} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 4.999999999099794 \cdot 10^{-24}:\\ \;\;\;\;\frac{e^{\frac{-\left|x\right|}{s}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-2 \cdot \log 2 + \frac{-0.25}{\frac{s \cdot s}{x \cdot x}}}}{s}\\ \end{array} \]

Alternative 6: 78.2% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-*l* 99.6%

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

   4. times-frac 99.6%

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

   5. associate-*r/ 99.6%

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

   6. associate-/l* 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. exp-neg 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around inf 51.3%

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

   1. associate-+r+ 51.3%

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

   2. distribute-lft1-in 51.3%

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

   3. metadata-eval 51.3%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0} \cdot \frac{\left|x\right|}{s}} \]

   4. mul0-lft 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\left(\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + 4\right) + \color{blue}{0}} \]

   5. associate-+r+ 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{{\left(\left|x\right|\right)}^{2}}{{s}^{2}} + \left(4 + 0\right)}} \]

   6. unpow2 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{{s}^{2}} + \left(4 + 0\right)} \]

   7. sqr-abs 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\frac{\color{blue}{x \cdot x}}{{s}^{2}} + \left(4 + 0\right)} \]

   8. unpow2 79.4%

      \[\leadsto \frac{\frac{1}{s}}{\frac{x \cdot x}{\color{blue}{s \cdot s}} + \left(4 + 0\right)} \]

   9. metadata-eval 79.4%

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

6. Simplified 79.4%

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

7. Final simplification 79.4%

   \[\leadsto \frac{\frac{1}{s}}{4 + \frac{x \cdot x}{s \cdot s}} \]

Alternative 7: 64.8% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -5.000000058430487e-8) (not (<= x 0.0007999999797903001)))
   (/ 1.0 (* x (/ x s)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -5.000000058430487e-8f) || !(x <= 0.0007999999797903001f)) {
		tmp = 1.0f / (x * (x / s));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-5.000000058430487e-8)) .or. (.not. (x <= 0.0007999999797903001e0))) then
        tmp = 1.0e0 / (x * (x / s))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-5.000000058430487e-8)) || !(x <= Float32(0.0007999999797903001)))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-5.000000058430487e-8)) || ~((x <= single(0.0007999999797903001))))
		tmp = single(1.0) / (x * (x / s));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8 or 7.9999998e-4 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 4.6%

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

      1. associate-+r+ 4.6%

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

      2. distribute-rgt-out 4.6%

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

      3. metadata-eval 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot \color{blue}{0} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      4. associate-+r+ 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      5. mul-1-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + \color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. unsub-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      7. fma-def 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\mathsf{fma}\left(2, \frac{{\left(\left|x\right|\right)}^{2}}{s}, 4 \cdot s\right)} - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      8. unpow2 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      9. sqr-abs 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{x \cdot x}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      10. *-commutative 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, \color{blue}{s \cdot 4}\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      11. unpow2 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      12. sqr-abs 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in x around inf 4.4%

      \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{2 \cdot \frac{{x}^{2}}{s}} - \frac{x \cdot x}{s}\right)} \]
   8. Step-by-step derivation

      1. unpow2 4.4%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(2 \cdot \frac{\color{blue}{x \cdot x}}{s} - \frac{x \cdot x}{s}\right)} \]

      2. associate-/l* 4.4%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(2 \cdot \color{blue}{\frac{x}{\frac{s}{x}}} - \frac{x \cdot x}{s}\right)} \]

      3. associate-/r/ 4.4%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(2 \cdot \color{blue}{\left(\frac{x}{s} \cdot x\right)} - \frac{x \cdot x}{s}\right)} \]

   9. Simplified 4.4%

      \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{2 \cdot \left(\frac{x}{s} \cdot x\right)} - \frac{x \cdot x}{s}\right)} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 4.4%

          \[\leadsto \color{blue}{1 \cdot \frac{1}{\left|x\right| \cdot 0 + \left(2 \cdot \left(\frac{x}{s} \cdot x\right) - \frac{x \cdot x}{s}\right)}} \]

       2. mul0-rgt 4.4%

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{0} + \left(2 \cdot \left(\frac{x}{s} \cdot x\right) - \frac{x \cdot x}{s}\right)} \]

       3. associate-+r- 4.4%

          \[\leadsto 1 \cdot \frac{1}{\color{blue}{\left(0 + 2 \cdot \left(\frac{x}{s} \cdot x\right)\right) - \frac{x \cdot x}{s}}} \]

       4. associate-*r* 4.4%

          \[\leadsto 1 \cdot \frac{1}{\left(0 + \color{blue}{\left(2 \cdot \frac{x}{s}\right) \cdot x}\right) - \frac{x \cdot x}{s}} \]

       5. associate-*l/ 4.4%

          \[\leadsto 1 \cdot \frac{1}{\left(0 + \left(2 \cdot \frac{x}{s}\right) \cdot x\right) - \color{blue}{\frac{x}{s} \cdot x}} \]

       6. *-commutative 4.4%

          \[\leadsto 1 \cdot \frac{1}{\left(0 + \left(2 \cdot \frac{x}{s}\right) \cdot x\right) - \color{blue}{x \cdot \frac{x}{s}}} \]

   11. Applied egg-rr 4.4%

       \[\leadsto \color{blue}{1 \cdot \frac{1}{\left(0 + \left(2 \cdot \frac{x}{s}\right) \cdot x\right) - x \cdot \frac{x}{s}}} \]
   12. Step-by-step derivation

       1. *-lft-identity 4.4%

          \[\leadsto \color{blue}{\frac{1}{\left(0 + \left(2 \cdot \frac{x}{s}\right) \cdot x\right) - x \cdot \frac{x}{s}}} \]

       2. +-lft-identity 4.4%

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

       3. *-lft-identity 4.4%

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

       4. associate-*l* 4.4%

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

       5. *-commutative 4.4%

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

       6. distribute-rgt-out-- 71.9%

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

       7. metadata-eval 71.9%

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

       8. *-rgt-identity 71.9%

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

   13. Simplified 71.9%

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

   if -5.00000006e-8 < x < 7.9999998e-4

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 52.2%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 8: 64.8% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -5.000000058430487e-8) (not (<= x 0.0007999999797903001)))
   (/ 1.0 (/ x (/ s x)))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -5.000000058430487e-8f) || !(x <= 0.0007999999797903001f)) {
		tmp = 1.0f / (x / (s / x));
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-5.000000058430487e-8)) .or. (.not. (x <= 0.0007999999797903001e0))) then
        tmp = 1.0e0 / (x / (s / x))
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-5.000000058430487e-8)) || !(x <= Float32(0.0007999999797903001)))
		tmp = Float32(Float32(1.0) / Float32(x / Float32(s / x)));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-5.000000058430487e-8)) || ~((x <= single(0.0007999999797903001))))
		tmp = single(1.0) / (x / (s / x));
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8 or 7.9999998e-4 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 4.6%

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

      1. associate-+r+ 4.6%

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

      2. distribute-rgt-out 4.6%

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

      3. metadata-eval 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot \color{blue}{0} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      4. associate-+r+ 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      5. mul-1-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + \color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. unsub-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      7. fma-def 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\mathsf{fma}\left(2, \frac{{\left(\left|x\right|\right)}^{2}}{s}, 4 \cdot s\right)} - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      8. unpow2 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      9. sqr-abs 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{x \cdot x}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      10. *-commutative 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, \color{blue}{s \cdot 4}\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      11. unpow2 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      12. sqr-abs 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in x around inf 70.1%

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

      1. unpow2 70.1%

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

   9. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   10. Step-by-step derivation

       1. div-inv 70.1%

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

   11. Applied egg-rr 70.1%

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

       1. div-inv 70.1%

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

       2. associate-/r* 70.1%

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

       3. clear-num 71.9%

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

   13. Applied egg-rr 71.9%

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

   if -5.00000006e-8 < x < 7.9999998e-4

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 52.2%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{\frac{x}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 9: 64.8% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -5.000000058430487e-8) (not (<= x 0.0007999999797903001)))
   (/ 1.0 (/ (* x x) s))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -5.000000058430487e-8f) || !(x <= 0.0007999999797903001f)) {
		tmp = 1.0f / ((x * x) / s);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-5.000000058430487e-8)) .or. (.not. (x <= 0.0007999999797903001e0))) then
        tmp = 1.0e0 / ((x * x) / s)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-5.000000058430487e-8)) || !(x <= Float32(0.0007999999797903001)))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-5.000000058430487e-8)) || ~((x <= single(0.0007999999797903001))))
		tmp = single(1.0) / ((x * x) / s);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8 or 7.9999998e-4 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 4.6%

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

      1. associate-+r+ 4.6%

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

      2. distribute-rgt-out 4.6%

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

      3. metadata-eval 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot \color{blue}{0} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      4. associate-+r+ 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      5. mul-1-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + \color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. unsub-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      7. fma-def 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\mathsf{fma}\left(2, \frac{{\left(\left|x\right|\right)}^{2}}{s}, 4 \cdot s\right)} - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      8. unpow2 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      9. sqr-abs 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{x \cdot x}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      10. *-commutative 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, \color{blue}{s \cdot 4}\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      11. unpow2 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      12. sqr-abs 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in x around inf 70.1%

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

      1. unpow2 70.1%

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

   9. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   10. Step-by-step derivation

       1. div-inv 70.1%

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

   11. Applied egg-rr 70.1%

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

       1. div-inv 70.1%

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

       2. clear-num 71.9%

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

   13. Applied egg-rr 71.9%

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

   if -5.00000006e-8 < x < 7.9999998e-4

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 52.2%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 10: 63.5% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -5.000000058430487e-8) (not (<= x 0.0007999999797903001)))
   (/ s (* x x))
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -5.000000058430487e-8f) || !(x <= 0.0007999999797903001f)) {
		tmp = s / (x * x);
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-5.000000058430487e-8)) .or. (.not. (x <= 0.0007999999797903001e0))) then
        tmp = s / (x * x)
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-5.000000058430487e-8)) || !(x <= Float32(0.0007999999797903001)))
		tmp = Float32(s / Float32(x * x));
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-5.000000058430487e-8)) || ~((x <= single(0.0007999999797903001))))
		tmp = s / (x * x);
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8 or 7.9999998e-4 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 4.6%

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

      1. associate-+r+ 4.6%

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

      2. distribute-rgt-out 4.6%

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

      3. metadata-eval 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot \color{blue}{0} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      4. associate-+r+ 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      5. mul-1-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + \color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. unsub-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      7. fma-def 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\mathsf{fma}\left(2, \frac{{\left(\left|x\right|\right)}^{2}}{s}, 4 \cdot s\right)} - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      8. unpow2 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      9. sqr-abs 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{x \cdot x}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      10. *-commutative 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, \color{blue}{s \cdot 4}\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      11. unpow2 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      12. sqr-abs 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in x around inf 70.1%

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

      1. unpow2 70.1%

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

   9. Simplified 70.1%

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

   if -5.00000006e-8 < x < 7.9999998e-4

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 52.2%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 11: 63.4% accurate, 66.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (or (<= x -5.000000058430487e-8) (not (<= x 0.0007999999797903001)))
   (/ (/ s x) x)
   (/ 0.25 s)))

C

float code(float x, float s) {
	float tmp;
	if ((x <= -5.000000058430487e-8f) || !(x <= 0.0007999999797903001f)) {
		tmp = (s / x) / x;
	} else {
		tmp = 0.25f / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x <= (-5.000000058430487e-8)) .or. (.not. (x <= 0.0007999999797903001e0))) then
        tmp = (s / x) / x
    else
        tmp = 0.25e0 / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if ((x <= Float32(-5.000000058430487e-8)) || !(x <= Float32(0.0007999999797903001)))
		tmp = Float32(Float32(s / x) / x);
	else
		tmp = Float32(Float32(0.25) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x <= single(-5.000000058430487e-8)) || ~((x <= single(0.0007999999797903001))))
		tmp = (s / x) / x;
	else
		tmp = single(0.25) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000006e-8 or 7.9999998e-4 < 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. *-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

      3. associate-/l* 99.9%

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

      4. distribute-frac-neg 99.9%

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

      5. exp-neg 99.9%

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

      6. associate-/r/ 99.9%

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

      7. /-rgt-identity 99.9%

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

      8. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in s around inf 4.6%

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

      1. associate-+r+ 4.6%

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

      2. distribute-rgt-out 4.6%

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

      3. metadata-eval 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot \color{blue}{0} + \left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + \left(4 \cdot s + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)\right)} \]

      4. associate-+r+ 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + -1 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      5. mul-1-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) + \color{blue}{\left(-\frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}\right)} \]

      6. unsub-neg 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \color{blue}{\left(\left(2 \cdot \frac{{\left(\left|x\right|\right)}^{2}}{s} + 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)}} \]

      7. fma-def 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\color{blue}{\mathsf{fma}\left(2, \frac{{\left(\left|x\right|\right)}^{2}}{s}, 4 \cdot s\right)} - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      8. unpow2 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      9. sqr-abs 4.6%

         \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{\color{blue}{x \cdot x}}{s}, 4 \cdot s\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      10. *-commutative 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, \color{blue}{s \cdot 4}\right) - \frac{{\left(\left|x\right|\right)}^{2}}{s}\right)} \]

      11. unpow2 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{\left|x\right| \cdot \left|x\right|}}{s}\right)} \]

      12. sqr-abs 4.6%

          \[\leadsto \frac{1}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{\color{blue}{x \cdot x}}{s}\right)} \]

   6. Simplified 4.6%

      \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \left(\mathsf{fma}\left(2, \frac{x \cdot x}{s}, s \cdot 4\right) - \frac{x \cdot x}{s}\right)}} \]

   7. Taylor expanded in x around inf 70.1%

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

      1. unpow2 70.1%

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

   9. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{s}{x \cdot x}} \]
   10. Step-by-step derivation

       1. div-inv 70.1%

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

   11. Applied egg-rr 70.1%

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

       1. div-inv 70.1%

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

       2. associate-/r* 70.1%

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

   13. Applied egg-rr 70.1%

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

   if -5.00000006e-8 < x < 7.9999998e-4

   1. Initial program 99.1%

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

      1. *-lft-identity 99.1%

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

      2. associate-*r/ 99.1%

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

      3. associate-*l* 99.0%

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

      4. times-frac 99.0%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. distribute-frac-neg 99.0%

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

      8. exp-neg 99.1%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in s around inf 52.2%

      \[\leadsto \color{blue}{\frac{0.25}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000058430487 \cdot 10^{-8} \lor \neg \left(x \leq 0.0007999999797903001\right):\\ \;\;\;\;\frac{\frac{s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s}\\ \end{array} \]

Alternative 12: 27.7% 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 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-*l* 99.6%

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

   4. times-frac 99.6%

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

   5. associate-*r/ 99.6%

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

   6. associate-/l* 99.6%

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

   7. distribute-frac-neg 99.6%

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

   8. exp-neg 99.6%

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

3. Simplified 99.7%

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

4. Taylor expanded in s around inf 23.4%

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

5. Final simplification 23.4%

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

Reproduce

herbie shell --seed 2023187 
(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))))))