Logistic distribution

Percentage Accurate: 99.4% → 99.4%

Time: 27.7s

Alternatives: 9

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.4% 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: 99.4% accurate, 1.2× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (fabs x) (- s)))) (fma s (pow E (/ x s)) s))))

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

   1. *-lft-identity 99.3%

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

   2. associate-*r/ 99.3%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.3%

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

   6. associate-/r/ 99.3%

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

   7. /-rgt-identity 99.3%

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

   8. associate-*l* 99.3%

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

3. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. exp-prod 67.4%

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

   3. add-sqr-sqrt 67.4%

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

   4. sqrt-unprod 67.4%

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

   5. sqr-neg 67.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 13.3%

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

   8. exp-prod 13.0%

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

   9. div-inv 13.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 99.4%

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

   12. frac-times 80.7%

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

   13. sqr-neg 80.7%

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

   14. sqr-neg 80.7%

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

   15. frac-times 99.4%

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

   16. sqrt-unprod -0.0%

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

5. Applied egg-rr 56.3%

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

6. Final simplification 56.3%

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

Alternative 2: 99.4% accurate, 1.5× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* (+ 1.0 (exp (/ (fabs x) (- s)))) (+ s (* s (pow E (/ x s)))))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

   1. *-lft-identity 99.3%

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

   2. associate-*r/ 99.3%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.3%

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

   6. associate-/r/ 99.3%

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

   7. /-rgt-identity 99.3%

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

   8. associate-*l* 99.3%

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

3. Simplified 99.4%

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

   1. fma-udef 99.4%

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

5. Applied egg-rr 56.2%

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

   1. *-un-lft-identity 56.2%

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

   2. pow-exp 56.2%

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

   3. e-exp-1 56.2%

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

7. Applied egg-rr 56.2%

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

8. Final simplification 56.2%

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

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ (exp (/ x s)) (+ (exp (/ (fabs x) (- s))) 2.0)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (exp((x / s)) + (exp((abs(x) / -s)) + 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

   1. div-inv 99.4%

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

   2. exp-prod 67.4%

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

   3. add-sqr-sqrt 67.4%

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

   4. sqrt-unprod 67.4%

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

   5. sqr-neg 67.4%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 13.3%

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

   8. exp-prod 13.0%

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

   9. div-inv 13.0%

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

   10. add-sqr-sqrt -0.0%

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

   11. sqrt-unprod 99.4%

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

   12. frac-times 80.7%

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

   13. sqr-neg 80.7%

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

   14. sqr-neg 80.7%

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

   15. frac-times 99.4%

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

   16. sqrt-unprod -0.0%

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

5. Applied egg-rr 56.0%

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

   1. expm1-log1p-u 54.8%

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

   2. expm1-udef 54.8%

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

   3. associate-/l/ 55.2%

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

   4. pow-to-exp 55.2%

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

   5. e-exp-1 55.2%

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

   6. add-log-exp 55.2%

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

   7. *-un-lft-identity 55.2%

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

7. Applied egg-rr 55.2%

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

   1. expm1-def 55.2%

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

   2. expm1-log1p 56.4%

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

   3. *-commutative 56.4%

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

   4. +-commutative 56.4%

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

9. Simplified 56.4%

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

10. Final simplification 56.4%

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

Alternative 4: 96.0% accurate, 3.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ 3.0 (exp (/ (fabs x) s))))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * (3.0e0 + exp((abs(x) / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 94.6%

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

5. Taylor expanded in s around 0 95.0%

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

6. Final simplification 95.0%

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

Alternative 5: 60.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{\mathsf{fma}\left(s, 4, x \cdot \frac{x}{s}\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 1.0 (fma s 4.0 (* x (/ x s)))))

C

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

Julia

x = abs(x)
function code(x, s)
	return Float32(Float32(1.0) / fma(s, Float32(4.0), Float32(x * Float32(x / s))))
end

Derivation

1. Initial program 99.3%

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

   1. *-lft-identity 99.3%

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

   2. associate-*r/ 99.3%

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

   3. associate-/l* 99.4%

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

   4. distribute-frac-neg 99.4%

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

   5. exp-neg 99.3%

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

   6. associate-/r/ 99.3%

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

   7. /-rgt-identity 99.3%

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

   8. associate-*l* 99.3%

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

3. Simplified 99.4%

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

4. Taylor expanded in s around -inf 37.1%

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

   1. associate-+r+ 37.1%

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

   2. distribute-rgt-out 37.1%

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

   3. metadata-eval 37.1%

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

   4. *-commutative 37.1%

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

   5. fma-def 37.1%

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

   6. mul-1-neg 37.1%

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

   7. distribute-rgt1-in 60.9%

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

   8. metadata-eval 60.9%

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

   9. associate-*r/ 60.9%

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

   10. mul-1-neg 60.9%

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

   11. remove-double-neg 60.9%

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

   12. unpow2 60.9%

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

   13. sqr-abs 60.9%

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

6. Simplified 60.9%

   \[\leadsto \frac{1}{\color{blue}{\left|x\right| \cdot 0 + \mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}} \]
7. Step-by-step derivation

   1. expm1-log1p-u 60.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)\right)} \]

   2. expm1-udef 73.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\left|x\right| \cdot 0 + \mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)} - 1} \]

   3. add-log-exp 68.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\log \left(e^{\left|x\right| \cdot 0 + \mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)}}\right)} - 1 \]

   4. mul0-rgt 68.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\log \left(e^{\color{blue}{0} + \mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)}\right)} - 1 \]

   5. exp-sum 68.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\log \color{blue}{\left(e^{0} \cdot e^{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)}}\right)} - 1 \]

   6. 1-exp 68.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\log \left(\color{blue}{1} \cdot e^{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)}\right)} - 1 \]

   7. *-un-lft-identity 68.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\log \color{blue}{\left(e^{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}\right)}}\right)} - 1 \]

   8. add-log-exp 73.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{\mathsf{fma}\left(s, 4, \frac{x \cdot x}{s}\right)}}\right)} - 1 \]

   9. associate-/l* 74.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(s, 4, \color{blue}{\frac{x}{\frac{s}{x}}}\right)}\right)} - 1 \]

8. Applied egg-rr 74.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(s, 4, \frac{x}{\frac{s}{x}}\right)}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 61.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(s, 4, \frac{x}{\frac{s}{x}}\right)}\right)\right)} \]

   2. expm1-log1p 62.3%

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

   3. associate-/r/ 62.3%

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

   4. *-commutative 62.3%

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

10. Simplified 62.3%

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

11. Final simplification 62.3%

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

Alternative 6: 95.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{s}}{e^{\frac{x}{s}} + 3} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (exp (/ x s)) 3.0)))

C

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

Fortran

NOTE: x should be positive before calling this function
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

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

4. Taylor expanded in s around inf 94.6%

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

   1. div-inv 94.6%

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

   2. exp-prod 69.1%

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

   3. add-sqr-sqrt 69.1%

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

   4. sqrt-unprod 69.1%

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

   5. sqr-neg 69.1%

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

   6. sqrt-unprod -0.0%

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

   7. add-sqr-sqrt 15.4%

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

   8. exp-prod 13.6%

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

   9. div-inv 13.6%

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

   10. distribute-frac-neg 13.6%

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

   11. rec-exp 13.6%

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

   12. *-un-lft-identity 13.6%

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

   13. rec-exp 13.6%

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

   14. distribute-frac-neg 13.6%

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

   15. add-sqr-sqrt -0.0%

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

   16. sqrt-unprod 88.3%

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

   17. sqr-neg 88.3%

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

   18. sqrt-unprod 94.6%

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

   19. add-sqr-sqrt 94.6%

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

6. Applied egg-rr 53.6%

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

   1. *-lft-identity 53.6%

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

8. Simplified 53.6%

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

9. Final simplification 53.6%

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

Alternative 7: 57.9% accurate, 67.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7000000840947678 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 3.7000000840947678e-9) (/ 0.25 s) (* s (/ 1.0 (* x x)))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 3.7000000840947678e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = s * (1.0f / (x * x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.7000000840947678e-9) then
        tmp = 0.25e0 / s
    else
        tmp = s * (1.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.7000000840947678e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s * Float32(Float32(1.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.7000000840947678e-9))
		tmp = single(0.25) / s;
	else
		tmp = s * (single(1.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 3.70000008e-9

   1. Initial program 99.0%

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

   2. Taylor expanded in s around inf 20.2%

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

   if 3.70000008e-9 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 42.9%

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

      1. associate-+r+ 42.9%

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

      2. distribute-rgt-out 42.9%

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

      3. metadata-eval 42.9%

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

      4. *-commutative 42.9%

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

      5. fma-def 42.9%

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

      6. mul-1-neg 42.9%

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

      7. distribute-rgt1-in 80.2%

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

      8. metadata-eval 80.2%

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

      9. associate-*r/ 80.2%

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

      10. mul-1-neg 80.2%

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

      11. remove-double-neg 80.2%

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

      12. unpow2 80.2%

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

      13. sqr-abs 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in x around inf 77.6%

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

      1. unpow2 77.6%

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

   9. Simplified 77.6%

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

       1. div-inv 77.6%

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

   11. Applied egg-rr 77.6%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7000000840947678 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{1}{x \cdot x}\\ \end{array} \]

Alternative 8: 57.9% accurate, 87.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7000000840947678 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s)
 :precision binary32
 (if (<= x 3.7000000840947678e-9) (/ 0.25 s) (/ s (* x x))))

C

x = abs(x);
float code(float x, float s) {
	float tmp;
	if (x <= 3.7000000840947678e-9f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 3.7000000840947678e-9) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

x = abs(x)
function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(3.7000000840947678e-9))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(3.7000000840947678e-9))
		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 < 3.70000008e-9

   1. Initial program 99.0%

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

   2. Taylor expanded in s around inf 20.2%

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

   if 3.70000008e-9 < x

   1. Initial program 100.0%

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

      1. *-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. distribute-frac-neg 100.0%

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

      5. exp-neg 100.0%

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

      6. associate-/r/ 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in s around -inf 42.9%

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

      1. associate-+r+ 42.9%

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

      2. distribute-rgt-out 42.9%

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

      3. metadata-eval 42.9%

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

      4. *-commutative 42.9%

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

      5. fma-def 42.9%

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

      6. mul-1-neg 42.9%

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

      7. distribute-rgt1-in 80.2%

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

      8. metadata-eval 80.2%

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

      9. associate-*r/ 80.2%

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

      10. mul-1-neg 80.2%

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

      11. remove-double-neg 80.2%

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

      12. unpow2 80.2%

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

      13. sqr-abs 80.2%

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

   6. Simplified 80.2%

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

   7. Taylor expanded in x around inf 77.6%

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

      1. unpow2 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7000000840947678 \cdot 10^{-9}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 9: 15.8% accurate, 206.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x s) :precision binary32 (/ 0.25 s))

C

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

Fortran

NOTE: x should be positive before calling this function
real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in s around inf 15.4%

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

3. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2023278 
(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))))))