Logistic distribution

Percentage Accurate: 99.5% → 99.6%

Time: 11.3s

Alternatives: 14

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(exp(Float32(abs(x) / Float32(-s))) + (Float32(exp(1)) ^ Float32(abs(x) / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(2.0) + (exp((abs(x) / -s)) + (single(2.71828182845904523536) ^ (abs(x) / s)))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. exp-1-e 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

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

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(2.0) + Float32(exp(Float32(abs(x) / Float32(-s))) + exp(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 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s + (s * (1.0f / expf((x / s))))) * (1.0f + 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 + (s * (1.0e0 / exp((x / s))))) * (1.0e0 + exp((abs(x) / s))))
end function

Julia

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

MATLAB

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

Derivation

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. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-/l* 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. exp-neg 99.7%

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

   6. associate-/r/ 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around 0 99.8%

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

   1. associate-*r* 99.8%

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

   2. distribute-lft-in 99.8%

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

   3. rem-exp-log 98.8%

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

   4. exp-sum 98.6%

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

   5. mul-1-neg 98.6%

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

   6. sub-neg 98.6%

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

   7. exp-diff 98.8%

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

   8. rem-exp-log 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

6. Simplified 99.8%

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

   1. add-exp-log 98.8%

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

   2. div-exp 98.6%

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

8. Applied egg-rr 98.6%

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

   1. exp-diff 98.8%

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

   2. add-exp-log 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 99.8%

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

   5. add-sqr-sqrt 99.8%

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

   6. add-sqr-sqrt 56.5%

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

   7. fabs-sqr 56.5%

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

   8. add-sqr-sqrt 97.8%

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

10. Applied egg-rr 97.8%

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

11. Final simplification 97.8%

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

Alternative 4: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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. *-lft-identity 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-/l* 99.7%

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

   4. distribute-frac-neg 99.7%

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

   5. exp-neg 99.7%

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

   6. associate-/r/ 99.7%

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

   7. /-rgt-identity 99.7%

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

   8. associate-*l* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around 0 99.8%

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

   1. associate-*r* 99.8%

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

   2. distribute-lft-in 99.8%

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

   3. rem-exp-log 98.8%

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

   4. exp-sum 98.6%

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

   5. mul-1-neg 98.6%

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

   6. sub-neg 98.6%

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

   7. exp-diff 98.8%

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

   8. rem-exp-log 99.8%

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

   9. *-rgt-identity 99.8%

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

   10. +-commutative 99.8%

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

6. Simplified 99.8%

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

   1. add-exp-log 98.8%

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

   2. div-exp 98.6%

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

8. Applied egg-rr 98.6%

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

   1. expm1-log1p-u 98.6%

      \[\leadsto \frac{1}{\left(s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\log s - \frac{\left|x\right|}{s}}\right)\right)}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]

   2. expm1-udef 86.3%

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

   3. exp-diff 86.3%

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

   4. add-exp-log 86.3%

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

   5. add-sqr-sqrt 86.3%

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

   6. add-sqr-sqrt 86.3%

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

   7. add-sqr-sqrt 48.3%

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

   8. fabs-sqr 48.3%

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

   9. add-sqr-sqrt 86.2%

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

10. Applied egg-rr 86.2%

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

    1. expm1-def 97.8%

       \[\leadsto \frac{1}{\left(s + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s}{e^{\frac{x}{s}}}\right)\right)}\right) \cdot \left(1 + e^{\frac{\left|x\right|}{s}}\right)} \]

    2. expm1-log1p 97.8%

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

12. Simplified 97.8%

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

13. Final simplification 97.8%

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

Alternative 5: 96.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left({e}^{\left(\frac{\left|x\right|}{s}\right)} + 3\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (pow E (/ (fabs x) s)) 3.0))))

C

float code(float x, float s) {
	return 1.0f / (s * (powf(((float) M_E), (fabsf(x) / s)) + 3.0f));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32((Float32(exp(1)) ^ Float32(abs(x) / s)) + Float32(3.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * ((single(2.71828182845904523536) ^ (abs(x) / s)) + single(3.0)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 97.2%

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

9. Taylor expanded in s around 0 97.2%

   \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{\left|x\right|}{s}}\right)}} \]
10. Step-by-step derivation

    1. *-un-lft-identity 99.8%

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

    2. exp-prod 99.9%

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

    3. exp-1-e 99.9%

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

11. Applied egg-rr 97.2%

    \[\leadsto \frac{1}{s \cdot \left(3 + \color{blue}{{e}^{\left(\frac{\left|x\right|}{s}\right)}}\right)} \]

12. Final simplification 97.2%

    \[\leadsto \frac{1}{s \cdot \left({e}^{\left(\frac{\left|x\right|}{s}\right)} + 3\right)} \]

Alternative 6: 60.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (exp (/ x s)) 3.0))))

C

float code(float x, float s) {
	return 1.0f / (s * (expf((x / s)) + 3.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((x / s)) + 3.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(exp(Float32(x / s)) + Float32(3.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (exp((x / s)) + single(3.0)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 97.2%

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

9. Taylor expanded in s around 0 97.2%

   \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{\left|x\right|}{s}}\right)}} \]
10. Step-by-step derivation

    1. distribute-lft-in 97.2%

       \[\leadsto \frac{1}{\color{blue}{s \cdot 3 + s \cdot e^{\frac{\left|x\right|}{s}}}} \]

    2. add-sqr-sqrt 97.2%

       \[\leadsto \frac{1}{s \cdot 3 + s \cdot e^{\color{blue}{\sqrt{\frac{\left|x\right|}{s}} \cdot \sqrt{\frac{\left|x\right|}{s}}}}} \]

    3. add-sqr-sqrt 97.2%

       \[\leadsto \frac{1}{s \cdot 3 + s \cdot e^{\color{blue}{\frac{\left|x\right|}{s}}}} \]

    4. add-sqr-sqrt 55.3%

       \[\leadsto \frac{1}{s \cdot 3 + s \cdot e^{\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{s}}} \]

    5. fabs-sqr 55.3%

       \[\leadsto \frac{1}{s \cdot 3 + s \cdot e^{\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{s}}} \]

    6. add-sqr-sqrt 66.1%

       \[\leadsto \frac{1}{s \cdot 3 + s \cdot e^{\frac{\color{blue}{x}}{s}}} \]

11. Applied egg-rr 66.1%

    \[\leadsto \frac{1}{\color{blue}{s \cdot 3 + s \cdot e^{\frac{x}{s}}}} \]
12. Step-by-step derivation

    1. distribute-lft-in 66.2%

       \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]

13. Simplified 66.2%

    \[\leadsto \frac{1}{\color{blue}{s \cdot \left(3 + e^{\frac{x}{s}}\right)}} \]

14. Final simplification 66.2%

    \[\leadsto \frac{1}{s \cdot \left(e^{\frac{x}{s}} + 3\right)} \]

Alternative 7: 82.0% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999996399175e-23)
   (/ 1.0 (* s (+ 4.0 (/ x (* s (/ s x))))))
   (/ 1.0 (* s (+ 4.0 (* (* x x) (/ 1.0 (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = 1.0f / (s * (4.0f + (x / (s * (s / x)))));
	} else {
		tmp = 1.0f / (s * (4.0f + ((x * x) * (1.0f / (s * 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 <= 1.9999999996399175e-23) then
        tmp = 1.0e0 / (s * (4.0e0 + (x / (s * (s / x)))))
    else
        tmp = 1.0e0 / (s * (4.0e0 + ((x * x) * (1.0e0 / (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(x / Float32(s * Float32(s / x))))));
	else
		tmp = Float32(Float32(1.0) / Float32(s * Float32(Float32(4.0) + Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = single(1.0) / (s * (single(4.0) + (x / (s * (s / x)))));
	else
		tmp = single(1.0) / (s * (single(4.0) + ((x * x) * (single(1.0) / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.7%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 97.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-udef 96.9%

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

   5. Applied egg-rr 96.9%

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

      1. expm1-def 97.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.9%

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

      3. associate-/l/ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in s around inf 60.4%

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

      1. associate-+r+ 60.4%

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

      2. distribute-lft1-in 60.4%

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

      3. metadata-eval 60.4%

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

      4. mul0-lft 80.6%

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

      5. associate-+r+ 80.6%

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

      6. metadata-eval 80.6%

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

      7. +-commutative 80.6%

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

      8. unpow2 80.6%

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

      9. sqr-abs 80.6%

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

      10. unpow2 80.6%

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

   10. Simplified 80.6%

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

       1. times-frac 78.8%

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

   12. Applied egg-rr 78.8%

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

       1. clear-num 78.8%

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

       2. frac-times 82.0%

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

       3. *-un-lft-identity 82.0%

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

   14. Applied egg-rr 82.0%

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

   if 2e-23 < x

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

      1. expm1-log1p-u 99.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-udef 98.7%

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

   5. Applied egg-rr 98.7%

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

      1. expm1-def 99.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-log1p 99.8%

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

      3. associate-/l/ 99.8%

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

      4. *-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in s around inf 47.3%

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

      1. associate-+r+ 47.3%

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

      2. distribute-lft1-in 47.3%

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

      3. metadata-eval 47.3%

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

      4. mul0-lft 80.3%

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

      5. associate-+r+ 80.3%

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

      6. metadata-eval 80.3%

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

      7. +-commutative 80.3%

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

      8. unpow2 80.3%

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

      9. sqr-abs 80.3%

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

      10. unpow2 80.3%

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

   10. Simplified 80.3%

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

       1. div-inv 81.2%

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

   12. Applied egg-rr 81.2%

       \[\leadsto \frac{1}{s \cdot \left(4 + \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{s \cdot s}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \frac{x}{s \cdot \frac{s}{x}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{s \cdot \left(4 + \left(x \cdot x\right) \cdot \frac{1}{s \cdot s}\right)}\\ \end{array} \]

Alternative 8: 77.1% accurate, 47.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (s * (4.0f + ((x / s) * (x / 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 / s) * (x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 54.7%

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

   1. associate-+r+ 54.7%

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

   2. distribute-lft1-in 54.7%

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

   3. metadata-eval 54.7%

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

   4. mul0-lft 80.5%

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

   5. associate-+r+ 80.5%

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

   6. metadata-eval 80.5%

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

   7. +-commutative 80.5%

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

   8. unpow2 80.5%

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

   9. sqr-abs 80.5%

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

   10. unpow2 80.5%

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

10. Simplified 80.5%

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

    1. times-frac 73.6%

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

12. Applied egg-rr 73.6%

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

13. Final simplification 73.6%

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

Alternative 9: 79.5% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 54.7%

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

   1. associate-+r+ 54.7%

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

   2. distribute-lft1-in 54.7%

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

   3. metadata-eval 54.7%

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

   4. mul0-lft 80.5%

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

   5. associate-+r+ 80.5%

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

   6. metadata-eval 80.5%

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

   7. +-commutative 80.5%

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

   8. unpow2 80.5%

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

   9. sqr-abs 80.5%

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

   10. unpow2 80.5%

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

10. Simplified 80.5%

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

    1. times-frac 73.6%

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

12. Applied egg-rr 73.6%

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

    1. clear-num 73.6%

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

    2. frac-times 77.2%

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

    3. *-un-lft-identity 77.2%

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

14. Applied egg-rr 77.2%

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

15. Final simplification 77.2%

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

Alternative 10: 78.5% accurate, 47.7× speedup

Math

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

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(1.0) / Float32(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 99.7%

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

3. Simplified 99.8%

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

   1. expm1-log1p-u 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-udef 97.7%

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

5. Applied egg-rr 97.7%

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

   1. expm1-def 98.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

   2. expm1-log1p 99.8%

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

   3. associate-/l/ 99.8%

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

   4. *-commutative 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in s around inf 54.7%

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

   1. associate-+r+ 54.7%

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

   2. distribute-lft1-in 54.7%

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

   3. metadata-eval 54.7%

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

   4. mul0-lft 80.5%

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

   5. associate-+r+ 80.5%

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

   6. metadata-eval 80.5%

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

   7. +-commutative 80.5%

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

   8. unpow2 80.5%

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

   9. sqr-abs 80.5%

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

   10. unpow2 80.5%

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

10. Simplified 80.5%

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

11. Final simplification 80.5%

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

Alternative 11: 45.7% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.00019999999494757503) (/ 0.25 s) (/ 1.0 (* x (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (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 (x <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (x * (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.00019999999494757503))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (x * (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   1. Initial program 99.6%

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

   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 38.6%

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

   if 1.99999995e-4 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 26.5%

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

      1. associate-+r+ 26.5%

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

      2. distribute-lft1-in 26.5%

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

      3. metadata-eval 26.5%

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

      4. mul0-lft 75.9%

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

      5. associate-+r+ 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

      8. unpow2 75.9%

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

      9. sqr-abs 75.9%

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

      10. unpow2 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in s around 0 66.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   12. Step-by-step derivation

       1. unpow2 66.6%

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

       2. associate-*r/ 66.6%

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

   13. Simplified 66.6%

       \[\leadsto \frac{1}{\color{blue}{x \cdot \frac{x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 12: 45.7% accurate, 67.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.00019999999494757503) (/ 0.25 s) (/ 1.0 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((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 (x <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.00019999999494757503))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   1. Initial program 99.6%

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

   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 38.6%

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

   if 1.99999995e-4 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 26.5%

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

      1. associate-+r+ 26.5%

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

      2. distribute-lft1-in 26.5%

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

      3. metadata-eval 26.5%

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

      4. mul0-lft 75.9%

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

      5. associate-+r+ 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

      8. unpow2 75.9%

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

      9. sqr-abs 75.9%

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

      10. unpow2 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in s around 0 66.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{2}}{s}}} \]
   12. Step-by-step derivation

       1. unpow2 66.6%

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

   13. Simplified 66.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot x}{s}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s}}\\ \end{array} \]

Alternative 13: 45.1% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.00019999999494757503) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.00019999999494757503f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.00019999999494757503e0) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.00019999999494757503))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999995e-4

   1. Initial program 99.6%

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

   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 38.6%

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

   if 1.99999995e-4 < x

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{s}}{e^{\frac{\left|x\right|}{s}} + \left(e^{\frac{\left|x\right|}{-s}} + 2\right)}\right)\right)} \]

      2. expm1-log1p 100.0%

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

      3. associate-/l/ 100.0%

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

      4. *-commutative 100.0%

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

      5. +-commutative 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+l+ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in s around inf 26.5%

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

      1. associate-+r+ 26.5%

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

      2. distribute-lft1-in 26.5%

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

      3. metadata-eval 26.5%

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

      4. mul0-lft 75.9%

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

      5. associate-+r+ 75.9%

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

      6. metadata-eval 75.9%

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

      7. +-commutative 75.9%

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

      8. unpow2 75.9%

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

      9. sqr-abs 75.9%

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

      10. unpow2 75.9%

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

   10. Simplified 75.9%

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

   11. Taylor expanded in s around 0 64.5%

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

       1. unpow2 64.5%

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

   13. Simplified 64.5%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 14: 26.6% 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 99.7%

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

3. Simplified 99.8%

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

4. Taylor expanded in s around inf 28.6%

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

5. Final simplification 28.6%

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

Reproduce

herbie shell --seed 2023258 
(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))))))