Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.2s

Alternatives: 11

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

   3. add-sqr-sqrt 51.5%

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

   4. sqrt-unprod 62.9%

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

   5. sqr-neg 62.9%

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

   6. sqrt-unprod 14.2%

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

   7. add-sqr-sqrt 26.7%

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

   8. add-cbrt-cube 26.7%

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

   9. pow1/3 26.7%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

   10. pow-flip 26.7%

       \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

3. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]
4. Step-by-step derivation

   1. add-exp-log 99.9%

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}\right)}} \]

   2. log-rec 100.0%

      \[\leadsto e^{\color{blue}{-\log \left(1 + {\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

   3. log1p-udef 100.0%

      \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

   4. add-exp-log 100.0%

      \[\leadsto e^{-\mathsf{log1p}\left(\color{blue}{e^{\log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}\right)} \]

   5. pow-pow 100.0%

      \[\leadsto e^{-\mathsf{log1p}\left(e^{\log \color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{\left(3 \cdot -0.3333333333333333\right)}\right)}}\right)} \]

   6. metadata-eval 100.0%

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

   7. log-pow 100.0%

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

   8. add-log-exp 100.0%

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

   9. neg-mul-1 100.0%

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

   10. distribute-neg-frac 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{\frac{x}{s} \cdot 3}\right)}^{-0.3333333333333333}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp (* (/ x s) 3.0)) -0.3333333333333333))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(((x / s) * 3.0f)), -0.3333333333333333f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp(((x / s) * 3.0e0)) ** (-0.3333333333333333e0)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(Float32(x / s) * Float32(3.0))) ^ Float32(-0.3333333333333333))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(((x / s) * single(3.0))) ^ single(-0.3333333333333333)));
end

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Step-by-step derivation

   1. distribute-frac-neg 99.9%

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

   2. exp-neg 99.9%

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

   3. add-sqr-sqrt 51.5%

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

   4. sqrt-unprod 62.9%

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

   5. sqr-neg 62.9%

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

   6. sqrt-unprod 14.2%

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

   7. add-sqr-sqrt 26.7%

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

   8. add-cbrt-cube 26.7%

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

   9. pow1/3 26.7%

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

   10. pow-flip 26.7%

       \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

3. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]
4. Step-by-step derivation

   1. expm1-log1p-u 99.9%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)\right)\right)}}^{-0.3333333333333333}} \]

   2. expm1-udef 99.5%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)} - 1\right)}}^{-0.3333333333333333}} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)} - 1\right)}}^{-0.3333333333333333}} \]
6. Step-by-step derivation

   1. expm1-def 99.9%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)\right)\right)}}^{-0.3333333333333333}} \]

   2. expm1-log1p 99.9%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}}^{-0.3333333333333333}} \]

   3. exp-prod 99.9%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\frac{x}{s} \cdot 3}\right)}}^{-0.3333333333333333}} \]

7. Simplified 99.9%

   \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\frac{x}{s} \cdot 3}\right)}}^{-0.3333333333333333}} \]

8. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + {\left(e^{\frac{x}{s} \cdot 3}\right)}^{-0.3333333333333333}} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 4: 60.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -20:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \left(x \cdot \frac{-3}{s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -20.0)
     0.5
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ 1.0 (+ 1.0 (+ 1.0 (* 0.3333333333333333 (* x (/ -3.0 s))))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -20.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.3333333333333333f * (x * (-3.0f / 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 = -x / s
    if (t_0 <= (-20.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.3333333333333333e0 * (x * ((-3.0e0) / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-20.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(x * Float32(Float32(-3.0) / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-20.0))
		tmp = single(0.5);
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.3333333333333333) * (x * (single(-3.0) / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -20 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. unsub-neg 54.8%

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

   4. Simplified 54.8%

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

      1. sub-neg 54.8%

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

      2. neg-mul-1 54.8%

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

      3. add-log-exp 95.7%

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

      4. log-pow 95.7%

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

      5. metadata-eval 95.7%

         \[\leadsto \frac{1}{2 + \log \left({\left(e^{\frac{x}{s}}\right)}^{\color{blue}{\left(3 \cdot -0.3333333333333333\right)}}\right)} \]

      6. pow-pow 95.7%

         \[\leadsto \frac{1}{2 + \log \color{blue}{\left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}} \]

      7. flip-+ 49.8%

         \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot 2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right) \cdot \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}{2 - \log \left({\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}\right)}}} \]

   6. Applied egg-rr 74.2%

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

   if 4.99999984e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. add-cbrt-cube 6.3%

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

      9. pow1/3 6.3%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

      10. pow-flip 6.3%

          \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

   3. Applied egg-rr 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   4. Taylor expanded in s around -inf 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-out 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot \left(-2 + -1\right)}}{s}\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   6. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]

   7. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(-3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot -3\right)}\right)} \]

      2. associate-*l/ 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\frac{x \cdot -3}{s}}\right)} \]

      3. associate-*r/ 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot \frac{-3}{s}\right)}\right)} \]

   9. Simplified 100.0%

      \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot \frac{-3}{s}\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \left(x \cdot \frac{-3}{s}\right)\right)}\\ \end{array} \]

Alternative 5: 49.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \left(x \cdot \frac{-3}{s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -20.0)
   0.5
   (/ 1.0 (+ 1.0 (+ 1.0 (* 0.3333333333333333 (* x (/ -3.0 s))))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.3333333333333333f * (x * (-3.0f / 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 / s) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.3333333333333333e0 * (x * ((-3.0e0) / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.3333333333333333) * Float32(x * Float32(Float32(-3.0) / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.3333333333333333) * (x * (single(-3.0) / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Step-by-step derivation

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

      3. add-sqr-sqrt 20.5%

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

      4. sqrt-unprod 41.0%

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

      5. sqr-neg 41.0%

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

      6. sqrt-unprod 23.3%

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

      7. add-sqr-sqrt 39.8%

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

      8. add-cbrt-cube 39.8%

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

      9. pow1/3 39.8%

         \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{0.3333333333333333}}}} \]

      10. pow-flip 39.8%

          \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(e^{\frac{-x}{s}} \cdot e^{\frac{-x}{s}}\right) \cdot e^{\frac{-x}{s}}\right)}^{\left(-0.3333333333333333\right)}}} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left({\left(e^{\frac{x}{s}}\right)}^{3}\right)}^{-0.3333333333333333}}} \]

   4. Taylor expanded in s around -inf 65.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{-2 \cdot x + -1 \cdot x}{s}\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-out 65.3%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{\color{blue}{x \cdot \left(-2 + -1\right)}}{s}\right)} \]

      2. metadata-eval 65.3%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \frac{x \cdot \color{blue}{-3}}{s}\right)} \]

   6. Simplified 65.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + 0.3333333333333333 \cdot \frac{x \cdot -3}{s}\right)}} \]

   7. Taylor expanded in x around 0 65.3%

      \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(-3 \cdot \frac{x}{s}\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 65.3%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(\frac{x}{s} \cdot -3\right)}\right)} \]

      2. associate-*l/ 65.3%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\frac{x \cdot -3}{s}}\right)} \]

      3. associate-*r/ 65.9%

         \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot \frac{-3}{s}\right)}\right)} \]

   9. Simplified 65.9%

      \[\leadsto \frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(x \cdot \frac{-3}{s}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.3333333333333333 \cdot \left(x \cdot \frac{-3}{s}\right)\right)}\\ \end{array} \]

Alternative 6: 49.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -20.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (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 ((-x / s) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

      \[\leadsto \color{blue}{0.5} \]

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. unsub-neg 64.8%

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

   4. Simplified 64.8%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 7: 48.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 1.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.8%

      \[\leadsto \color{blue}{0.5} \]

   if 1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. unsub-neg 45.1%

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

   4. Simplified 45.1%

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

   5. Taylor expanded in x around inf 45.1%

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

      1. mul-1-neg 45.1%

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

      2. distribute-frac-neg 45.1%

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

   7. Simplified 45.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 8: 46.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (* s (/ 1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = s * (1.0f / x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = s * (1.0e0 / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = s * (single(1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   4. Simplified 54.0%

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

   5. Taylor expanded in x around inf 49.9%

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

      1. associate-*r/ 49.9%

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

      2. neg-mul-1 49.9%

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

   7. Simplified 49.9%

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

      1. div-inv 49.9%

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

      2. *-commutative 49.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.7%

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

      5. sqr-neg 63.7%

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

      6. sqrt-unprod 49.9%

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

      7. add-sqr-sqrt 49.9%

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

   9. Applied egg-rr 49.9%

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

   if -9.99999972e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.4%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 47.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   4. Simplified 54.0%

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

   5. Taylor expanded in x around inf 49.9%

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

      1. associate-*r/ 49.9%

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

      2. neg-mul-1 49.9%

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

   7. Simplified 49.9%

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

      1. div-inv 49.9%

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

      2. *-commutative 49.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.7%

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

      5. sqr-neg 63.7%

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

      6. sqrt-unprod 49.9%

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

      7. add-sqr-sqrt 49.9%

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

   9. Applied egg-rr 49.9%

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

       1. associate-/r/ 54.0%

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

   11. Applied egg-rr 54.0%

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

   if -9.99999972e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.4%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 46.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999717180685e-10) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999717180685e-10f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999717180685e-10)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999717180685e-10))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999717180685e-10))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999972e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   4. Simplified 54.0%

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

   5. Taylor expanded in x around inf 49.9%

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

      1. associate-*r/ 49.9%

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

      2. neg-mul-1 49.9%

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

   7. Simplified 49.9%

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

      1. expm1-log1p-u 49.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-s}{x}\right)\right)} \]

      2. expm1-udef 96.2%

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

      3. frac-2neg 96.2%

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

      4. frac-2neg 96.2%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 96.2%

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

      7. sqr-neg 96.2%

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

      8. sqrt-unprod 96.2%

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

      9. add-sqr-sqrt 96.2%

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

   9. Applied egg-rr 96.2%

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

       1. expm1-def 49.9%

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

       2. expm1-log1p 49.9%

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

   11. Simplified 49.9%

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

   if -9.99999972e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.4%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 11: 34.3% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 35.5%

   \[\leadsto \color{blue}{0.5} \]

3. Final simplification 35.5%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))