Logistic distribution

Percentage Accurate: 99.5% → 99.4%

Time: 9.6s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.5% 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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (exp((abs(x) / -s)) + (exp((abs(x) / s)) + single(2.0)));
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. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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* 99.6%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. add-exp-log 98.6%

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

   3. prod-exp 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 19.8%

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

   7. sqr-neg 19.8%

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

   8. sqrt-unprod 21.7%

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

   9. add-sqr-sqrt 21.7%

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

   10. prod-exp 21.7%

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

   11. add-exp-log 22.3%

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

   12. *-rgt-identity 22.3%

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

   13. distribute-lft-in 22.3%

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

   14. +-commutative 22.3%

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

   15. *-commutative 22.3%

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

5. Applied egg-rr 62.1%

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

   1. remove-double-neg 27.1%

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

   2. distribute-frac-neg 27.1%

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

   3. frac-2neg 27.1%

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

   4. exp-neg 27.1%

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

   5. div-inv 27.1%

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

   6. exp-prod 26.2%

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

   7. add-sqr-sqrt 26.2%

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

   8. sqrt-unprod 26.2%

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

   9. sqr-neg 26.2%

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

   10. sqrt-unprod 19.2%

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

   11. add-sqr-sqrt 82.1%

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

   12. exp-prod 95.3%

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

   13. div-inv 95.3%

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

   14. add-sqr-sqrt 48.3%

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

   15. fabs-sqr 48.3%

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

   16. add-sqr-sqrt 60.9%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 58.3%

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

   19. sqr-neg 58.3%

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

   20. sqrt-unprod 61.5%

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

   21. add-sqr-sqrt 61.5%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * (single(1.0) + exp((x / s)))) * (single(1.0) + 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* 99.6%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. add-exp-log 98.6%

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

   3. prod-exp 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 19.8%

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

   7. sqr-neg 19.8%

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

   8. sqrt-unprod 21.7%

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

   9. add-sqr-sqrt 21.7%

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

   10. prod-exp 21.7%

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

   11. add-exp-log 22.3%

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

   12. *-rgt-identity 22.3%

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

   13. distribute-lft-in 22.3%

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

   14. +-commutative 22.3%

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

   15. *-commutative 22.3%

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

5. Applied egg-rr 62.1%

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

   1. remove-double-neg 27.1%

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

   2. distribute-frac-neg 27.1%

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

   3. frac-2neg 27.1%

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

   4. exp-neg 27.1%

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

   5. div-inv 27.1%

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

   6. exp-prod 26.2%

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

   7. add-sqr-sqrt 26.2%

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

   8. sqrt-unprod 26.2%

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

   9. sqr-neg 26.2%

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

   10. sqrt-unprod 19.2%

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

   11. add-sqr-sqrt 82.1%

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

   12. exp-prod 95.3%

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

   13. div-inv 95.3%

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

   14. add-sqr-sqrt 48.3%

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

   15. fabs-sqr 48.3%

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

   16. add-sqr-sqrt 60.9%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 58.3%

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

   19. sqr-neg 58.3%

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

   20. sqrt-unprod 61.5%

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

   21. add-sqr-sqrt 61.5%

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

7. Applied egg-rr 99.7%

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

   1. rec-exp 61.5%

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

   2. distribute-neg-frac 61.5%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 4: 62.7% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / ((s * (single(1.0) + exp((x / s)))) * (single(1.0) + (single(1.0) / (single(1.0) + (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* 99.6%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. add-exp-log 98.6%

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

   3. prod-exp 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 19.8%

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

   7. sqr-neg 19.8%

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

   8. sqrt-unprod 21.7%

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

   9. add-sqr-sqrt 21.7%

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

   10. prod-exp 21.7%

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

   11. add-exp-log 22.3%

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

   12. *-rgt-identity 22.3%

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

   13. distribute-lft-in 22.3%

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

   14. +-commutative 22.3%

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

   15. *-commutative 22.3%

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

5. Applied egg-rr 62.1%

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

   1. remove-double-neg 27.1%

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

   2. distribute-frac-neg 27.1%

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

   3. frac-2neg 27.1%

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

   4. exp-neg 27.1%

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

   5. div-inv 27.1%

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

   6. exp-prod 26.2%

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

   7. add-sqr-sqrt 26.2%

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

   8. sqrt-unprod 26.2%

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

   9. sqr-neg 26.2%

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

   10. sqrt-unprod 19.2%

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

   11. add-sqr-sqrt 82.1%

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

   12. exp-prod 95.3%

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

   13. div-inv 95.3%

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

   14. add-sqr-sqrt 48.3%

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

   15. fabs-sqr 48.3%

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

   16. add-sqr-sqrt 60.9%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 58.3%

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

   19. sqr-neg 58.3%

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

   20. sqrt-unprod 61.5%

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

   21. add-sqr-sqrt 61.5%

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

7. Applied egg-rr 99.7%

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

8. Taylor expanded in x around 0 61.5%

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

9. Final simplification 61.5%

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

Alternative 5: 61.1% accurate, 5.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (1.0f / s) / (2.0f + (2.0f * expf((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 + (2.0e0 * exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

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 97.6%

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

5. Applied egg-rr 58.8%

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

   1. expm1-def 58.8%

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

   2. expm1-log1p 59.7%

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

   3. associate-+r+ 59.7%

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

   4. count-2 59.7%

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

7. Simplified 59.7%

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

8. Final simplification 59.7%

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

Alternative 6: 55.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\\ \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \left(s \cdot \left(1 + t_0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ 1.0 (+ (/ x s) (* 0.5 (/ (* x x) (* s s)))))))
   (if (<= x 1.5000000786160286e-23)
     (/ 0.25 s)
     (if (<= x 200000.0)
       (/
        1.0
        (* (- (- 2.0 (/ x s)) (/ (* x (* x -0.5)) (* s s))) (* s (+ 1.0 t_0))))
       (/ (/ 1.0 s) (+ 2.0 (* 2.0 t_0)))))))

C

float code(float x, float s) {
	float t_0 = 1.0f + ((x / s) + (0.5f * ((x * x) / (s * s))));
	float tmp;
	if (x <= 1.5000000786160286e-23f) {
		tmp = 0.25f / s;
	} else if (x <= 200000.0f) {
		tmp = 1.0f / (((2.0f - (x / s)) - ((x * (x * -0.5f)) / (s * s))) * (s * (1.0f + t_0)));
	} else {
		tmp = (1.0f / s) / (2.0f + (2.0f * t_0));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 + ((x / s) + (0.5e0 * ((x * x) / (s * s))))
    if (x <= 1.5000000786160286e-23) then
        tmp = 0.25e0 / s
    else if (x <= 200000.0e0) then
        tmp = 1.0e0 / (((2.0e0 - (x / s)) - ((x * (x * (-0.5e0))) / (s * s))) * (s * (1.0e0 + t_0)))
    else
        tmp = (1.0e0 / s) / (2.0e0 + (2.0e0 * t_0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))))
	tmp = Float32(0.0)
	if (x <= Float32(1.5000000786160286e-23))
		tmp = Float32(Float32(0.25) / s);
	elseif (x <= Float32(200000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(2.0) - Float32(x / s)) - Float32(Float32(x * Float32(x * Float32(-0.5))) / Float32(s * s))) * Float32(s * Float32(Float32(1.0) + t_0))));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(Float32(2.0) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) + ((x / s) + (single(0.5) * ((x * x) / (s * s))));
	tmp = single(0.0);
	if (x <= single(1.5000000786160286e-23))
		tmp = single(0.25) / s;
	elseif (x <= single(200000.0))
		tmp = single(1.0) / (((single(2.0) - (x / s)) - ((x * (x * single(-0.5))) / (s * s))) * (s * (single(1.0) + t_0)));
	else
		tmp = (single(1.0) / s) / (single(2.0) + (single(2.0) * t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 1.50000008e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in s around inf 34.1%

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

   if 1.50000008e-23 < x < 2e5

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*r/ 99.4%

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

      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 99.4%

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

      6. associate-/r/ 99.4%

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

      7. /-rgt-identity 99.4%

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

      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%

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

      2. add-exp-log 98.3%

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

      3. prod-exp 98.1%

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

      4. frac-2neg 98.1%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 17.8%

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

      7. sqr-neg 17.8%

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

      8. sqrt-unprod 17.8%

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

      9. add-sqr-sqrt 17.8%

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

      10. prod-exp 17.8%

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

      11. add-exp-log 17.8%

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

      12. *-rgt-identity 17.8%

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

      13. distribute-lft-in 17.8%

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

      14. +-commutative 17.8%

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

      15. *-commutative 17.8%

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

   5. Applied egg-rr 99.3%

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

      1. remove-double-neg 22.7%

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

      2. distribute-frac-neg 22.7%

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

      3. frac-2neg 22.7%

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

      4. exp-neg 22.7%

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

      5. div-inv 22.7%

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

      6. exp-prod 19.7%

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

      7. add-sqr-sqrt 19.7%

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

      8. sqrt-unprod 19.7%

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

      9. sqr-neg 19.7%

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

      10. sqrt-unprod 16.1%

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

      11. add-sqr-sqrt 73.2%

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

      12. exp-prod 89.0%

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

      13. div-inv 89.0%

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

      14. add-sqr-sqrt 89.0%

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

      15. fabs-sqr 89.0%

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

      16. add-sqr-sqrt 89.0%

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

      17. add-sqr-sqrt -0.0%

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

      18. sqrt-unprod 22.7%

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

      19. sqr-neg 22.7%

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

      20. sqrt-unprod 22.7%

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

      21. add-sqr-sqrt 22.7%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in s around inf 91.0%

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

      1. +-commutative 91.0%

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

      2. mul-1-neg 91.0%

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

      3. unsub-neg 91.0%

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

      4. neg-mul-1 91.0%

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

      5. unsub-neg 91.0%

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

      6. distribute-rgt-out 91.0%

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

      7. unpow2 91.0%

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

      8. metadata-eval 91.0%

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

      9. associate-*l* 91.0%

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

      10. unpow2 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in x around 0 68.4%

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

       1. unpow2 62.5%

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

       2. unpow2 62.5%

          \[\leadsto \frac{\frac{1}{s}}{2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 2} \]

   13. Simplified 68.4%

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

   if 2e5 < 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%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-+r+ 100.0%

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

      4. count-2 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 97.3%

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

      1. unpow2 97.3%

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

      2. unpow2 97.3%

         \[\leadsto \frac{\frac{1}{s}}{2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 2} \]

   10. Simplified 97.3%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \left(s \cdot \left(1 + \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 7: 53.6% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \frac{x}{s}\\ t_1 := x + s \cdot 2\\ \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 199999995904:\\ \;\;\;\;\frac{1}{\left(t_0 - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 2.0 (/ x s))) (t_1 (+ x (* s 2.0))))
   (if (<= x 1.5000000786160286e-23)
     (/ 0.25 s)
     (if (<= x 199999995904.0)
       (/ 1.0 (* (- t_0 (/ (* x (* x -0.5)) (* s s))) t_1))
       (/ 1.0 (* t_0 t_1))))))

C

float code(float x, float s) {
	float t_0 = 2.0f - (x / s);
	float t_1 = x + (s * 2.0f);
	float tmp;
	if (x <= 1.5000000786160286e-23f) {
		tmp = 0.25f / s;
	} else if (x <= 199999995904.0f) {
		tmp = 1.0f / ((t_0 - ((x * (x * -0.5f)) / (s * s))) * t_1);
	} else {
		tmp = 1.0f / (t_0 * t_1);
	}
	return tmp;
}

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
    real(4) :: tmp
    t_0 = 2.0e0 - (x / s)
    t_1 = x + (s * 2.0e0)
    if (x <= 1.5000000786160286e-23) then
        tmp = 0.25e0 / s
    else if (x <= 199999995904.0e0) then
        tmp = 1.0e0 / ((t_0 - ((x * (x * (-0.5e0))) / (s * s))) * t_1)
    else
        tmp = 1.0e0 / (t_0 * t_1)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(2.0) - Float32(x / s))
	t_1 = Float32(x + Float32(s * Float32(2.0)))
	tmp = Float32(0.0)
	if (x <= Float32(1.5000000786160286e-23))
		tmp = Float32(Float32(0.25) / s);
	elseif (x <= Float32(199999995904.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(t_0 - Float32(Float32(x * Float32(x * Float32(-0.5))) / Float32(s * s))) * t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(t_0 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(2.0) - (x / s);
	t_1 = x + (s * single(2.0));
	tmp = single(0.0);
	if (x <= single(1.5000000786160286e-23))
		tmp = single(0.25) / s;
	elseif (x <= single(199999995904.0))
		tmp = single(1.0) / ((t_0 - ((x * (x * single(-0.5))) / (s * s))) * t_1);
	else
		tmp = single(1.0) / (t_0 * t_1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 1.50000008e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in s around inf 34.1%

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

   if 1.50000008e-23 < x < 199999996000

   1. Initial program 99.5%

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

      1. *-lft-identity 99.5%

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

      2. associate-*r/ 99.5%

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

      3. associate-/l* 99.5%

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

      4. distribute-frac-neg 99.5%

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

      5. exp-neg 99.5%

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

      6. associate-/r/ 99.5%

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

      7. /-rgt-identity 99.5%

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

      8. associate-*l* 99.4%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

      2. add-exp-log 98.6%

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

      3. prod-exp 98.4%

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

      4. frac-2neg 98.4%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 15.6%

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

      7. sqr-neg 15.6%

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

      8. sqrt-unprod 15.6%

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

      9. add-sqr-sqrt 15.6%

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

      10. prod-exp 15.6%

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

      11. add-exp-log 15.6%

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

      12. *-rgt-identity 15.6%

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

      13. distribute-lft-in 15.6%

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

      14. +-commutative 15.6%

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

      15. *-commutative 15.6%

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

   5. Applied egg-rr 99.4%

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

      1. remove-double-neg 20.3%

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

      2. distribute-frac-neg 20.3%

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

      3. frac-2neg 20.3%

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

      4. exp-neg 20.3%

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

      5. div-inv 20.3%

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

      6. exp-prod 17.7%

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

      7. add-sqr-sqrt 17.7%

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

      8. sqrt-unprod 17.7%

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

      9. sqr-neg 17.7%

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

      10. sqrt-unprod 13.5%

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

      11. add-sqr-sqrt 77.5%

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

      12. exp-prod 90.8%

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

      13. div-inv 90.8%

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

      14. add-sqr-sqrt 90.8%

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

      15. fabs-sqr 90.8%

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

      16. add-sqr-sqrt 90.8%

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

      17. add-sqr-sqrt -0.0%

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

      18. sqrt-unprod 20.3%

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

      19. sqr-neg 20.3%

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

      20. sqrt-unprod 20.3%

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

      21. add-sqr-sqrt 20.3%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in s around inf 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. neg-mul-1 86.4%

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

      5. unsub-neg 86.4%

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

      6. distribute-rgt-out 86.4%

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

      7. unpow2 86.4%

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

      8. metadata-eval 86.4%

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

      9. associate-*l* 86.4%

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

      10. unpow2 86.4%

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

   10. Simplified 86.4%

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

   11. Taylor expanded in x around 0 63.6%

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

   if 199999996000 < 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. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. add-exp-log 100.0%

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

      3. prod-exp 100.0%

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

      4. frac-2neg 100.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. sqrt-unprod 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. prod-exp 4.4%

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

      11. add-exp-log 4.4%

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

      12. *-rgt-identity 4.4%

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

      13. distribute-lft-in 4.4%

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

      14. +-commutative 4.4%

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

      15. *-commutative 4.4%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 12.3%

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

      1. remove-double-neg 12.3%

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

      2. distribute-frac-neg 12.3%

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

      3. frac-2neg 12.3%

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

      4. exp-neg 12.3%

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

      5. div-inv 12.3%

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

      6. exp-prod 12.3%

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

      7. add-sqr-sqrt 12.3%

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

      8. sqrt-unprod 12.3%

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

      9. sqr-neg 12.3%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. exp-prod 100.0%

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

      13. div-inv 100.0%

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

      14. add-sqr-sqrt 100.0%

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

      15. fabs-sqr 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. add-sqr-sqrt -0.0%

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

      18. sqrt-unprod 12.3%

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

      19. sqr-neg 12.3%

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

      20. sqrt-unprod 12.3%

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

      21. add-sqr-sqrt 12.3%

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

   8. Applied egg-rr 12.3%

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

      1. rec-exp 12.3%

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

      2. distribute-neg-frac 12.3%

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

   10. Simplified 12.3%

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

   11. Taylor expanded in x around 0 98.5%

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

       1. mul-1-neg 98.5%

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

       2. sub-neg 98.5%

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

   13. Simplified 98.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 199999995904:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \left(x + s \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) \cdot \left(x + s \cdot 2\right)}\\ \end{array} \]

Alternative 8: 52.6% accurate, 24.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 - \frac{x}{s}\\ \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 4000000:\\ \;\;\;\;\frac{1}{\left(t_0 - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \left(s \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0 \cdot \left(x + s \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 2.0 (/ x s))))
   (if (<= x 1.5000000786160286e-23)
     (/ 0.25 s)
     (if (<= x 4000000.0)
       (/ 1.0 (* (- t_0 (/ (* x (* x -0.5)) (* s s))) (* s 2.0)))
       (/ 1.0 (* t_0 (+ x (* s 2.0))))))))

C

float code(float x, float s) {
	float t_0 = 2.0f - (x / s);
	float tmp;
	if (x <= 1.5000000786160286e-23f) {
		tmp = 0.25f / s;
	} else if (x <= 4000000.0f) {
		tmp = 1.0f / ((t_0 - ((x * (x * -0.5f)) / (s * s))) * (s * 2.0f));
	} else {
		tmp = 1.0f / (t_0 * (x + (s * 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 2.0e0 - (x / s)
    if (x <= 1.5000000786160286e-23) then
        tmp = 0.25e0 / s
    else if (x <= 4000000.0e0) then
        tmp = 1.0e0 / ((t_0 - ((x * (x * (-0.5e0))) / (s * s))) * (s * 2.0e0))
    else
        tmp = 1.0e0 / (t_0 * (x + (s * 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(2.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (x <= Float32(1.5000000786160286e-23))
		tmp = Float32(Float32(0.25) / s);
	elseif (x <= Float32(4000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(t_0 - Float32(Float32(x * Float32(x * Float32(-0.5))) / Float32(s * s))) * Float32(s * Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(t_0 * Float32(x + Float32(s * Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(2.0) - (x / s);
	tmp = single(0.0);
	if (x <= single(1.5000000786160286e-23))
		tmp = single(0.25) / s;
	elseif (x <= single(4000000.0))
		tmp = single(1.0) / ((t_0 - ((x * (x * single(-0.5))) / (s * s))) * (s * single(2.0)));
	else
		tmp = single(1.0) / (t_0 * (x + (s * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < 1.50000008e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in s around inf 34.1%

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

   if 1.50000008e-23 < x < 4e6

   1. Initial program 99.4%

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

      1. *-lft-identity 99.4%

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

      2. associate-*r/ 99.4%

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

      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 99.4%

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

      6. associate-/r/ 99.4%

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

      7. /-rgt-identity 99.4%

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

      8. associate-*l* 99.4%

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

   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%

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

      2. add-exp-log 98.4%

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

      3. prod-exp 98.1%

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

      4. frac-2neg 98.1%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 17.2%

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

      7. sqr-neg 17.2%

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

      8. sqrt-unprod 17.2%

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

      9. add-sqr-sqrt 17.2%

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

      10. prod-exp 17.2%

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

      11. add-exp-log 17.2%

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

      12. *-rgt-identity 17.2%

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

      13. distribute-lft-in 17.2%

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

      14. +-commutative 17.2%

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

      15. *-commutative 17.2%

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

   5. Applied egg-rr 99.4%

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

      1. remove-double-neg 22.0%

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

      2. distribute-frac-neg 22.0%

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

      3. frac-2neg 22.0%

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

      4. exp-neg 22.0%

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

      5. div-inv 22.0%

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

      6. exp-prod 19.1%

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

      7. add-sqr-sqrt 19.1%

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

      8. sqrt-unprod 19.1%

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

      9. sqr-neg 19.1%

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

      10. sqrt-unprod 15.3%

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

      11. add-sqr-sqrt 74.4%

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

      12. exp-prod 89.5%

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

      13. div-inv 89.5%

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

      14. add-sqr-sqrt 89.5%

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

      15. fabs-sqr 89.5%

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

      16. add-sqr-sqrt 89.5%

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

      17. add-sqr-sqrt -0.0%

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

      18. sqrt-unprod 22.0%

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

      19. sqr-neg 22.0%

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

      20. sqrt-unprod 22.0%

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

      21. add-sqr-sqrt 22.0%

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

   7. Applied egg-rr 99.4%

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

   8. Taylor expanded in s around inf 89.1%

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

      1. +-commutative 89.1%

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

      2. mul-1-neg 89.1%

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

      3. unsub-neg 89.1%

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

      4. neg-mul-1 89.1%

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

      5. unsub-neg 89.1%

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

      6. distribute-rgt-out 89.1%

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

      7. unpow2 89.1%

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

      8. metadata-eval 89.1%

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

      9. associate-*l* 89.1%

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

      10. unpow2 89.1%

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

   10. Simplified 89.1%

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

   11. Taylor expanded in x around 0 59.6%

       \[\leadsto \frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \color{blue}{\left(2 \cdot s\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 59.6%

          \[\leadsto \frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \color{blue}{\left(s \cdot 2\right)}} \]

   13. Simplified 59.6%

       \[\leadsto \frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \color{blue}{\left(s \cdot 2\right)}} \]

   if 4e6 < 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. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. add-exp-log 100.0%

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

      3. prod-exp 100.0%

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

      4. frac-2neg 100.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. sqrt-unprod 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. prod-exp 4.4%

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

      11. add-exp-log 4.4%

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

      12. *-rgt-identity 4.4%

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

      13. distribute-lft-in 4.4%

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

      14. +-commutative 4.4%

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

      15. *-commutative 4.4%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 11.9%

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

      1. remove-double-neg 11.9%

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

      2. distribute-frac-neg 11.9%

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

      3. frac-2neg 11.9%

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

      4. exp-neg 11.9%

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

      5. div-inv 11.9%

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

      6. exp-prod 11.9%

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

      7. add-sqr-sqrt 11.9%

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

      8. sqrt-unprod 11.9%

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

      9. sqr-neg 11.9%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. exp-prod 100.0%

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

      13. div-inv 100.0%

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

      14. add-sqr-sqrt 100.0%

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

      15. fabs-sqr 100.0%

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

      16. add-sqr-sqrt 100.0%

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

      17. add-sqr-sqrt -0.0%

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

      18. sqrt-unprod 11.9%

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

      19. sqr-neg 11.9%

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

      20. sqrt-unprod 11.9%

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

      21. add-sqr-sqrt 11.9%

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

   8. Applied egg-rr 11.9%

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

      1. rec-exp 11.9%

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

      2. distribute-neg-frac 11.9%

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

   10. Simplified 11.9%

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

   11. Taylor expanded in x around 0 96.0%

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

       1. mul-1-neg 96.0%

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

       2. sub-neg 96.0%

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

   13. Simplified 96.0%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{elif}\;x \leq 4000000:\\ \;\;\;\;\frac{1}{\left(\left(2 - \frac{x}{s}\right) - \frac{x \cdot \left(x \cdot -0.5\right)}{s \cdot s}\right) \cdot \left(s \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) \cdot \left(x + s \cdot 2\right)}\\ \end{array} \]

Alternative 9: 54.1% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.5000000786160286e-23)
   (/ 0.25 s)
   (/
    (/ 1.0 s)
    (+ 2.0 (* 2.0 (+ 1.0 (+ (/ x s) (* 0.5 (/ (* x x) (* s s))))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.5000000786160286e-23f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / (2.0f + (2.0f * (1.0f + ((x / s) + (0.5f * ((x * x) / (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 <= 1.5000000786160286e-23) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / (2.0e0 + (2.0e0 * (1.0e0 + ((x / s) + (0.5e0 * ((x * x) / (s * s)))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.5000000786160286e-23))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(2.0) + Float32(Float32(2.0) * Float32(Float32(1.0) + Float32(Float32(x / s) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.5000000786160286e-23))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / (single(2.0) + (single(2.0) * (single(1.0) + ((x / s) + (single(0.5) * ((x * x) / (s * s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.50000008e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in s around inf 34.1%

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

   if 1.50000008e-23 < x

   1. Initial program 99.8%

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

   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 99.3%

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

   5. Applied egg-rr 96.0%

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

      1. expm1-def 96.0%

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

      2. expm1-log1p 96.0%

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

      3. associate-+r+ 96.0%

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

      4. count-2 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in x around 0 83.8%

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

      1. unpow2 83.8%

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

      2. unpow2 83.8%

         \[\leadsto \frac{\frac{1}{s}}{2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}\right)\right) + 2} \]

   10. Simplified 83.8%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5000000786160286 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{2 + 2 \cdot \left(1 + \left(\frac{x}{s} + 0.5 \cdot \frac{x \cdot x}{s \cdot s}\right)\right)}\\ \end{array} \]

Alternative 10: 64.4% accurate, 47.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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* 99.6%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. add-exp-log 98.6%

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

   3. prod-exp 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 19.8%

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

   7. sqr-neg 19.8%

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

   8. sqrt-unprod 21.7%

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

   9. add-sqr-sqrt 21.7%

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

   10. prod-exp 21.7%

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

   11. add-exp-log 22.3%

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

   12. *-rgt-identity 22.3%

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

   13. distribute-lft-in 22.3%

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

   14. +-commutative 22.3%

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

   15. *-commutative 22.3%

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

5. Applied egg-rr 62.1%

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

6. Taylor expanded in x around 0 27.1%

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

   1. remove-double-neg 27.1%

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

   2. distribute-frac-neg 27.1%

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

   3. frac-2neg 27.1%

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

   4. exp-neg 27.1%

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

   5. div-inv 27.1%

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

   6. exp-prod 26.2%

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

   7. add-sqr-sqrt 26.2%

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

   8. sqrt-unprod 26.2%

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

   9. sqr-neg 26.2%

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

   10. sqrt-unprod 19.2%

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

   11. add-sqr-sqrt 82.1%

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

   12. exp-prod 95.3%

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

   13. div-inv 95.3%

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

   14. add-sqr-sqrt 48.3%

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

   15. fabs-sqr 48.3%

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

   16. add-sqr-sqrt 60.9%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 58.3%

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

   19. sqr-neg 58.3%

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

   20. sqrt-unprod 61.5%

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

   21. add-sqr-sqrt 61.5%

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

8. Applied egg-rr 61.5%

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

   1. rec-exp 61.5%

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

   2. distribute-neg-frac 61.5%

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

10. Simplified 61.5%

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

11. Taylor expanded in x around 0 63.6%

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

    1. mul-1-neg 63.6%

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

    2. sub-neg 63.6%

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

13. Simplified 63.6%

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

14. Final simplification 63.6%

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

Alternative 11: 29.0% accurate, 55.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} - \frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.003000000026077032) (/ 0.25 s) (- (/ 0.5 x) (/ s (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.003000000026077032f) {
		tmp = 0.25f / s;
	} else {
		tmp = (0.5f / x) - (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 <= 0.003000000026077032e0) then
        tmp = 0.25e0 / s
    else
        tmp = (0.5e0 / x) - (s / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.003000000026077032))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(0.5) / x) - Float32(s / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.003000000026077032))
		tmp = single(0.25) / s;
	else
		tmp = (single(0.5) / x) - (s / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00300000003

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf 33.8%

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

   if 0.00300000003 < 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. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. add-exp-log 100.0%

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

      3. prod-exp 100.0%

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

      4. frac-2neg 100.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. sqrt-unprod 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. prod-exp 4.4%

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

      11. add-exp-log 4.4%

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

      12. *-rgt-identity 4.4%

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

      13. distribute-lft-in 4.4%

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

      14. +-commutative 4.4%

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

      15. *-commutative 4.4%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 10.7%

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

   7. Taylor expanded in s around inf 10.9%

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

   8. Taylor expanded in s around 0 10.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{s}{{x}^{2}} + 0.5 \cdot \frac{1}{x}} \]
   9. Step-by-step derivation

      1. +-commutative 10.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + -1 \cdot \frac{s}{{x}^{2}}} \]

      2. mul-1-neg 10.9%

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

      3. unsub-neg 10.9%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} - \frac{s}{{x}^{2}}} \]

      4. associate-*r/ 10.9%

         \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} - \frac{s}{{x}^{2}} \]

      5. metadata-eval 10.9%

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

      6. unpow2 10.9%

         \[\leadsto \frac{0.5}{x} - \frac{s}{\color{blue}{x \cdot x}} \]

   10. Simplified 10.9%

       \[\leadsto \color{blue}{\frac{0.5}{x} - \frac{s}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x} - \frac{s}{x \cdot x}\\ \end{array} \]

Alternative 12: 51.0% accurate, 56.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{2 \cdot \frac{x}{s} + 4} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ (* 2.0 (/ x s)) 4.0)))

C

float code(float x, float s) {
	return (1.0f / s) / ((2.0f * (x / s)) + 4.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / ((2.0e0 * (x / s)) + 4.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(Float32(2.0) * Float32(x / s)) + Float32(4.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((single(2.0) * (x / s)) + single(4.0));
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 97.6%

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

5. Applied egg-rr 58.8%

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

   1. expm1-def 58.8%

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

   2. expm1-log1p 59.7%

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

   3. associate-+r+ 59.7%

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

   4. count-2 59.7%

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

7. Simplified 59.7%

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

8. Taylor expanded in x around 0 51.3%

   \[\leadsto \frac{\frac{1}{s}}{\color{blue}{2 \cdot \frac{x}{s} + 4}} \]

9. Final simplification 51.3%

   \[\leadsto \frac{\frac{1}{s}}{2 \cdot \frac{x}{s} + 4} \]

Alternative 13: 29.4% accurate, 68.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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* 99.6%

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

3. Simplified 99.7%

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

   1. fma-udef 99.7%

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

   2. add-exp-log 98.6%

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

   3. prod-exp 98.5%

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

   4. frac-2neg 98.5%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 19.8%

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

   7. sqr-neg 19.8%

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

   8. sqrt-unprod 21.7%

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

   9. add-sqr-sqrt 21.7%

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

   10. prod-exp 21.7%

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

   11. add-exp-log 22.3%

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

   12. *-rgt-identity 22.3%

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

   13. distribute-lft-in 22.3%

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

   14. +-commutative 22.3%

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

   15. *-commutative 22.3%

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

5. Applied egg-rr 62.1%

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

6. Taylor expanded in x around 0 27.1%

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

7. Taylor expanded in s around inf 26.3%

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

8. Final simplification 26.3%

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

Alternative 14: 29.0% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.003000000026077032) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.003000000026077032f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.5f / 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 <= 0.003000000026077032e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.5e0 / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.003000000026077032))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.003000000026077032))
		tmp = single(0.25) / s;
	else
		tmp = single(0.5) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00300000003

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf 33.8%

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

   if 0.00300000003 < 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. Step-by-step derivation

      1. fma-udef 100.0%

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

      2. add-exp-log 100.0%

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

      3. prod-exp 100.0%

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

      4. frac-2neg 100.0%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.4%

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

      7. sqr-neg 4.4%

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

      8. sqrt-unprod 4.4%

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

      9. add-sqr-sqrt 4.4%

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

      10. prod-exp 4.4%

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

      11. add-exp-log 4.4%

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

      12. *-rgt-identity 4.4%

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

      13. distribute-lft-in 4.4%

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

      14. +-commutative 4.4%

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

      15. *-commutative 4.4%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 10.7%

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

   7. Taylor expanded in s around inf 10.9%

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

   8. Taylor expanded in s around 0 10.9%

      \[\leadsto \color{blue}{\frac{0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.003000000026077032:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 15: 27.4% 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%

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

2. Taylor expanded in s around inf 24.4%

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

3. Final simplification 24.4%

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

Reproduce

herbie shell --seed 2023240 
(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))))))