Logistic distribution

Percentage Accurate: 99.6% → 99.6%

Time: 8.8s

Alternatives: 14

Speedup: 1.2×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 97.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. remove-double-neg 59.1%

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

   2. distribute-frac-neg 59.1%

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

   3. frac-2neg 59.1%

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

   4. exp-neg 59.1%

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

   5. div-inv 59.1%

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

   6. exp-prod 57.1%

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

   7. add-sqr-sqrt 57.1%

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

   8. sqrt-unprod 57.1%

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

   9. sqr-neg 57.1%

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

   10. sqrt-unprod 30.1%

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

   11. add-sqr-sqrt 86.5%

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

   12. exp-prod 96.3%

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

   13. div-inv 96.3%

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

   14. add-sqr-sqrt 45.7%

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

   15. fabs-sqr 45.7%

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

   16. add-sqr-sqrt 56.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 96.0%

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

   19. sqr-neg 96.0%

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

   20. sqrt-unprod 99.5%

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

   21. add-sqr-sqrt 99.5%

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

5. Applied egg-rr 98.2%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 3: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

Derivation

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 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 59.1%

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

   1. remove-double-neg 59.1%

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

   2. distribute-frac-neg 59.1%

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

   3. frac-2neg 59.1%

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

   4. exp-neg 59.1%

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

   5. div-inv 59.1%

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

   6. exp-prod 57.1%

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

   7. add-sqr-sqrt 57.1%

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

   8. sqrt-unprod 57.1%

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

   9. sqr-neg 57.1%

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

   10. sqrt-unprod 30.1%

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

   11. add-sqr-sqrt 86.5%

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

   12. exp-prod 96.3%

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

   13. div-inv 96.3%

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

   14. add-sqr-sqrt 45.7%

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

   15. fabs-sqr 45.7%

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

   16. add-sqr-sqrt 56.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 96.0%

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

   19. sqr-neg 96.0%

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

   20. sqrt-unprod 99.5%

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

   21. add-sqr-sqrt 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.6% accurate, 2.9× speedup

Math

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

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(s * Float32(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 99.4%

   \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]
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 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 59.1%

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

   1. remove-double-neg 59.1%

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

   2. distribute-frac-neg 59.1%

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

   3. frac-2neg 59.1%

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

   4. exp-neg 59.1%

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

   5. div-inv 59.1%

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

   6. exp-prod 57.1%

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

   7. add-sqr-sqrt 57.1%

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

   8. sqrt-unprod 57.1%

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

   9. sqr-neg 57.1%

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

   10. sqrt-unprod 30.1%

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

   11. add-sqr-sqrt 86.5%

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

   12. exp-prod 96.3%

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

   13. div-inv 96.3%

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

   14. add-sqr-sqrt 45.7%

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

   15. fabs-sqr 45.7%

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

   16. add-sqr-sqrt 56.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 96.0%

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

   19. sqr-neg 96.0%

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

   20. sqrt-unprod 99.5%

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

   21. add-sqr-sqrt 99.5%

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

7. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in s around 0 99.4%

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

11. Taylor expanded in x around inf 99.4%

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

    1. associate-*r* 99.4%

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

    2. +-commutative 99.4%

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

    3. *-commutative 99.4%

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

    4. mul-1-neg 99.4%

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

    5. distribute-frac-neg 99.4%

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

    6. associate-*l* 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 5: 99.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 59.1%

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

   1. remove-double-neg 59.1%

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

   2. distribute-frac-neg 59.1%

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

   3. frac-2neg 59.1%

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

   4. exp-neg 59.1%

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

   5. div-inv 59.1%

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

   6. exp-prod 57.1%

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

   7. add-sqr-sqrt 57.1%

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

   8. sqrt-unprod 57.1%

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

   9. sqr-neg 57.1%

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

   10. sqrt-unprod 30.1%

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

   11. add-sqr-sqrt 86.5%

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

   12. exp-prod 96.3%

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

   13. div-inv 96.3%

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

   14. add-sqr-sqrt 45.7%

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

   15. fabs-sqr 45.7%

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

   16. add-sqr-sqrt 56.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 96.0%

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

   19. sqr-neg 96.0%

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

   20. sqrt-unprod 99.5%

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

   21. add-sqr-sqrt 99.5%

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

7. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 61.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((s + (s * 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 + (s * 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(s * 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 + (s * exp((x / s)))) * (single(1.0) + (single(1.0) / (single(1.0) + (x / s)))));
end

Derivation

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 99.5%

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

   1. fma-udef 99.5%

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

5. Applied egg-rr 59.1%

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

   1. remove-double-neg 59.1%

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

   2. distribute-frac-neg 59.1%

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

   3. frac-2neg 59.1%

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

   4. exp-neg 59.1%

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

   5. div-inv 59.1%

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

   6. exp-prod 57.1%

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

   7. add-sqr-sqrt 57.1%

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

   8. sqrt-unprod 57.1%

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

   9. sqr-neg 57.1%

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

   10. sqrt-unprod 30.1%

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

   11. add-sqr-sqrt 86.5%

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

   12. exp-prod 96.3%

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

   13. div-inv 96.3%

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

   14. add-sqr-sqrt 45.7%

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

   15. fabs-sqr 45.7%

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

   16. add-sqr-sqrt 56.0%

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

   17. add-sqr-sqrt -0.0%

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

   18. sqrt-unprod 96.0%

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

   19. sqr-neg 96.0%

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

   20. sqrt-unprod 99.5%

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

   21. add-sqr-sqrt 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in x around 0 57.9%

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

9. Final simplification 57.9%

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

Alternative 7: 60.1% accurate, 5.6× speedup

Math

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

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(1.0) / Float32(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 99.4%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 97.6%

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

   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 55.1%

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

   1. expm1-def 55.1%

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

   2. expm1-log1p 56.4%

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

   3. associate-+r+ 56.4%

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

   4. count-2 56.4%

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

7. Simplified 56.4%

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

8. Taylor expanded in s around 0 56.4%

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

9. Final simplification 56.4%

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

Alternative 8: 65.8% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \frac{x}{s} + 4\\ \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot \frac{s}{x}} + t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{t_0 + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (+ (* 2.0 (/ x s)) 4.0)))
   (if (<= x 1.9999999996399175e-23)
     (/ (/ 1.0 s) (+ (/ x (* s (/ s x))) t_0))
     (/ (/ 1.0 s) (+ t_0 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = (2.0f * (x / s)) + 4.0f;
	float tmp;
	if (x <= 1.9999999996399175e-23f) {
		tmp = (1.0f / s) / ((x / (s * (s / x))) + t_0);
	} else {
		tmp = (1.0f / s) / (t_0 + ((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) :: t_0
    real(4) :: tmp
    t_0 = (2.0e0 * (x / s)) + 4.0e0
    if (x <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / s) / ((x / (s * (s / x))) + t_0)
    else
        tmp = (1.0e0 / s) / (t_0 + ((x * x) / (s * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(Float32(2.0) * Float32(x / s)) + Float32(4.0))
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x / Float32(s * Float32(s / x))) + t_0));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(t_0 + Float32(Float32(x * x) / Float32(s * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = (single(2.0) * (x / s)) + single(4.0);
	tmp = single(0.0);
	if (x <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / s) / ((x / (s * (s / x))) + t_0);
	else
		tmp = (single(1.0) / s) / (t_0 + ((x * x) / (s * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2e-23

   1. Initial program 99.1%

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

   3. Simplified 99.3%

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

      1. expm1-log1p-u 96.7%

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

      2. expm1-udef 96.7%

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

   5. Applied egg-rr 31.4%

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

      1. expm1-def 31.5%

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

      2. expm1-log1p 33.4%

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

      3. associate-+r+ 33.4%

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

      4. count-2 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in x around 0 54.5%

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

      1. +-commutative 54.5%

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

      2. +-commutative 54.5%

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

      3. associate-+l+ 54.5%

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

      4. unpow2 54.5%

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

      5. unpow2 54.5%

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

      6. times-frac 53.4%

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

      7. +-commutative 53.4%

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

      8. *-commutative 53.4%

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

      9. fma-def 53.4%

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

   10. Simplified 53.4%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s} + \mathsf{fma}\left(\frac{x}{s}, 2, 4\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 53.4%

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

   12. Applied egg-rr 53.4%

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

       1. clear-num 53.4%

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

       2. frac-times 58.1%

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

       3. *-un-lft-identity 58.1%

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

   14. Applied egg-rr 58.1%

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

   if 2e-23 < x

   1. Initial program 99.7%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.0%

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

      2. expm1-udef 99.0%

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

   5. Applied egg-rr 95.3%

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

      1. expm1-def 95.3%

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

      2. expm1-log1p 95.5%

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

      3. associate-+r+ 95.5%

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

      4. count-2 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in x around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. +-commutative 79.9%

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

      3. associate-+l+ 79.9%

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

      4. unpow2 79.9%

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

      5. unpow2 79.9%

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

      6. times-frac 71.9%

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

      7. +-commutative 71.9%

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

      8. *-commutative 71.9%

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

      9. fma-def 71.9%

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

   10. Simplified 71.9%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s} + \mathsf{fma}\left(\frac{x}{s}, 2, 4\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 71.9%

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

   12. Applied egg-rr 71.9%

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

       1. frac-times 79.9%

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

   14. Applied egg-rr 79.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x}{s \cdot \frac{s}{x}} + \left(2 \cdot \frac{x}{s} + 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\left(2 \cdot \frac{x}{s} + 4\right) + \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 64.0% accurate, 32.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return (1.0f / s) / ((x / (s * (s / x))) + ((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) / ((x / (s * (s / x))) + ((2.0e0 * (x / s)) + 4.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 97.6%

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

   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 55.1%

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

   1. expm1-def 55.1%

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

   2. expm1-log1p 56.4%

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

   3. associate-+r+ 56.4%

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

   4. count-2 56.4%

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

7. Simplified 56.4%

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

8. Taylor expanded in x around 0 63.9%

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

   1. +-commutative 63.9%

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

   2. +-commutative 63.9%

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

   3. associate-+l+ 63.9%

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

   4. unpow2 63.9%

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

   5. unpow2 63.9%

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

   6. times-frac 60.3%

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

   7. +-commutative 60.3%

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

   8. *-commutative 60.3%

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

   9. fma-def 60.3%

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

10. Simplified 60.3%

    \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s} + \mathsf{fma}\left(\frac{x}{s}, 2, 4\right)}} \]
11. Step-by-step derivation

    1. fma-udef 60.3%

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

12. Applied egg-rr 60.3%

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

    1. clear-num 60.3%

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

    2. frac-times 64.3%

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

    3. *-un-lft-identity 64.3%

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

14. Applied egg-rr 64.3%

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

15. Final simplification 64.3%

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

Alternative 10: 76.2% accurate, 36.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 1.400000027358074 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= s 1.400000027358074e-24)
   (/ (/ 1.0 s) (/ (* x x) (* s s)))
   (/ (/ 1.0 s) (+ 4.0 (* (/ x s) (+ 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (s <= 1.400000027358074e-24f) {
		tmp = (1.0f / s) / ((x * x) / (s * s));
	} else {
		tmp = (1.0f / s) / (4.0f + ((x / s) * (2.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (s <= 1.400000027358074e-24) then
        tmp = (1.0e0 / s) / ((x * x) / (s * s))
    else
        tmp = (1.0e0 / s) / (4.0e0 + ((x / s) * (2.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (s <= Float32(1.400000027358074e-24))
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(x * x) / Float32(s * s)));
	else
		tmp = Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(Float32(2.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (s <= single(1.400000027358074e-24))
		tmp = (single(1.0) / s) / ((x * x) / (s * s));
	else
		tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * (single(2.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 1.40000003e-24

   1. Initial program 98.7%

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

   3. Simplified 98.8%

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

      1. expm1-log1p-u 97.6%

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

      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 46.7%

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

      1. expm1-def 46.7%

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

      2. expm1-log1p 47.1%

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

      3. associate-+r+ 47.1%

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

      4. count-2 47.1%

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

   7. Simplified 47.1%

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

   8. Taylor expanded in x around 0 59.1%

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

      1. +-commutative 59.1%

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

      2. +-commutative 59.1%

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

      3. associate-+l+ 59.1%

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

      4. unpow2 59.1%

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

      5. unpow2 59.1%

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

      6. times-frac 47.7%

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

      7. +-commutative 47.7%

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

      8. *-commutative 47.7%

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

      9. fma-def 47.7%

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

   10. Simplified 47.7%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s} + \mathsf{fma}\left(\frac{x}{s}, 2, 4\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 47.7%

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

   12. Applied egg-rr 47.7%

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

   13. Taylor expanded in x around inf 77.4%

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

       1. unpow2 77.4%

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

       2. unpow2 77.4%

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

   15. Simplified 77.4%

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

   if 1.40000003e-24 < s

   1. Initial program 99.7%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 97.6%

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

      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 59.9%

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

      1. expm1-def 60.0%

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

      2. expm1-log1p 61.7%

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

      3. associate-+r+ 61.7%

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

      4. count-2 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in x around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. +-commutative 66.6%

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

      3. associate-+l+ 66.6%

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

      4. unpow2 66.6%

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

      5. unpow2 66.6%

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

      6. times-frac 67.4%

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

      7. +-commutative 67.4%

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

      8. *-commutative 67.4%

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

      9. fma-def 67.4%

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

   10. Simplified 67.4%

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

   11. Taylor expanded in x around 0 66.6%

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

       1. +-commutative 66.6%

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

       2. unpow2 66.6%

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

       3. unpow2 66.6%

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

       4. times-frac 67.4%

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

       5. unpow2 67.4%

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

       6. *-commutative 67.4%

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

       7. associate-+l+ 67.4%

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

       8. unpow2 67.4%

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

       9. times-frac 66.6%

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

       10. unpow2 66.6%

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

       11. unpow2 66.6%

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

       12. *-commutative 66.6%

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

       13. +-commutative 66.6%

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

       14. unpow2 66.6%

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

       15. unpow2 66.6%

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

       16. times-frac 67.4%

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

       17. distribute-rgt-out 79.7%

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

   13. Simplified 79.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.400000027358074 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot \left(2 + \frac{x}{s}\right)}\\ \end{array} \]

Alternative 11: 50.9% accurate, 47.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.999999936531045e-20)
   (/ 0.25 s)
   (/ (/ 1.0 s) (/ (* x x) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999936531045e-20f) {
		tmp = 0.25f / s;
	} else {
		tmp = (1.0f / s) / ((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 <= 1.999999936531045e-20) then
        tmp = 0.25e0 / s
    else
        tmp = (1.0e0 / s) / ((x * x) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999936531045e-20))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / s) / 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(1.999999936531045e-20))
		tmp = single(0.25) / s;
	else
		tmp = (single(1.0) / s) / ((x * x) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999994e-20

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 33.7%

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

   if 1.99999994e-20 < x

   1. Initial program 99.7%

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

   3. Simplified 99.9%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 99.2%

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

   5. Applied egg-rr 95.7%

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

      1. expm1-def 95.7%

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

      2. expm1-log1p 95.8%

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

      3. associate-+r+ 95.8%

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

      4. count-2 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in x around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. +-commutative 80.4%

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

      3. associate-+l+ 80.4%

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

      4. unpow2 80.4%

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

      5. unpow2 80.4%

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

      6. times-frac 73.0%

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

      7. +-commutative 73.0%

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

      8. *-commutative 73.0%

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

      9. fma-def 73.0%

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

   10. Simplified 73.0%

       \[\leadsto \frac{\frac{1}{s}}{\color{blue}{\frac{x}{s} \cdot \frac{x}{s} + \mathsf{fma}\left(\frac{x}{s}, 2, 4\right)}} \]
   11. Step-by-step derivation

       1. fma-udef 73.0%

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

   12. Applied egg-rr 73.0%

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

   13. Taylor expanded in x around inf 70.2%

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

       1. unpow2 70.2%

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

       2. unpow2 70.2%

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

   15. Simplified 70.2%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{s}}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 12: 45.2% accurate, 87.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0005000000237487257:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0005000000237487257) (/ 0.25 s) (/ s (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0005000000237487257f) {
		tmp = 0.25f / s;
	} else {
		tmp = s / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.0005000000237487257e0) then
        tmp = 0.25e0 / s
    else
        tmp = s / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0005000000237487257))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(s / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0005000000237487257))
		tmp = single(0.25) / s;
	else
		tmp = s / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000024e-4

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 35.3%

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

   if 5.00000024e-4 < x

   1. Initial program 99.9%

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

   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 99.9%

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

   5. Applied egg-rr 98.8%

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

      1. expm1-def 98.8%

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

      2. expm1-log1p 98.8%

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

      3. associate-+r+ 98.8%

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

      4. count-2 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 82.1%

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

      1. +-commutative 82.1%

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

      2. +-commutative 82.1%

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

      3. associate-+l+ 82.1%

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

      4. unpow2 82.1%

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

      5. unpow2 82.1%

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

      6. times-frac 82.1%

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

      7. +-commutative 82.1%

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

      8. *-commutative 82.1%

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

      9. fma-def 82.1%

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

   10. Simplified 82.1%

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

   11. Taylor expanded in s around 0 65.8%

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

       1. unpow2 65.8%

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

   13. Simplified 65.8%

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

4. Final simplification 42.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005000000237487257:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x \cdot x}\\ \end{array} \]

Alternative 13: 28.6% accurate, 121.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.09600000083446503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.09600000083446503) (/ 0.25 s) (/ 0.5 x)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.09600000083446503f) {
		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 <= 0.09600000083446503e0) 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(0.09600000083446503))
		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(0.09600000083446503))
		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 < 0.0960000008

   1. Initial program 99.2%

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

   2. Taylor expanded in s around inf 34.9%

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

   if 0.0960000008 < x

   1. Initial program 99.9%

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

   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 99.9%

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

   5. Applied egg-rr 98.8%

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

      1. expm1-def 98.8%

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

      2. expm1-log1p 98.8%

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

      3. associate-+r+ 98.8%

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

      4. count-2 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 45.7%

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

   9. Taylor expanded in s around 0 11.2%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.09600000083446503:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 14: 26.9% 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 99.4%

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

2. Taylor expanded in s around inf 28.2%

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

3. Final simplification 28.2%

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

Reproduce

herbie shell --seed 2023257 
(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))))))