Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

Alternatives: 14

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: 99.8% 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 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 82.5%

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

   3. neg-mul-1 82.5%

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

   4. exp-prod 82.5%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. add-exp-log 99.7%

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

   2. log-rec 99.7%

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

   3. log1p-udef 99.8%

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

   4. pow-exp 99.8%

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

   5. neg-mul-1 99.8%

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

   6. distribute-neg-frac 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{\frac{s}{2}}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp -0.5) (/ x (/ s 2.0))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-0.5f), (x / (s / 2.0f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((-0.5e0)) ** (x / (s / 2.0e0))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-0.5)) ^ Float32(x / Float32(s / Float32(2.0))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-0.5)) ^ (x / (s / single(2.0)))));
end

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 82.5%

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

   3. neg-mul-1 82.5%

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

   4. exp-prod 82.5%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. unpow-prod-down 99.7%

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

5. Applied egg-rr 99.7%

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

   1. pow-sqr 99.7%

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

   2. associate-*r/ 99.7%

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

   3. *-commutative 99.7%

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

7. Simplified 99.7%

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

   1. expm1-log1p-u 99.7%

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

   2. expm1-udef 98.2%

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

   3. +-commutative 98.2%

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

   4. pow1/2 98.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\color{blue}{\left({\left(e^{-1}\right)}^{0.5}\right)}}^{\left(\frac{x \cdot 2}{s}\right)} + 1}\right)} - 1 \]

   5. pow-exp 98.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\color{blue}{\left(e^{-1 \cdot 0.5}\right)}}^{\left(\frac{x \cdot 2}{s}\right)} + 1}\right)} - 1 \]

   6. metadata-eval 98.2%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{{\left(e^{\color{blue}{-0.5}}\right)}^{\left(\frac{x \cdot 2}{s}\right)} + 1}\right)} - 1 \]

9. Applied egg-rr 98.2%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{{\left(e^{-0.5}\right)}^{\left(\frac{x \cdot 2}{s}\right)} + 1}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 99.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(e^{-0.5}\right)}^{\left(\frac{x \cdot 2}{s}\right)} + 1}\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. +-commutative 99.7%

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

    4. associate-/l* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 82.5%

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

   3. neg-mul-1 82.5%

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

   4. exp-prod 82.5%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 5: 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 99.7%

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

2. Final simplification 99.7%

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

Alternative 6: 59.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     0.5
     (if (<= t_0 1.9999999867631625e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ 2.0 (/ x s))))
       (/ 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 if (t_0 <= 1.9999999867631625e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (2.0f + (x / s)));
	} 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 if (t_0 <= 1.9999999867631625e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (2.0e0 + (x / s)))
    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);
	elseif (t_0 <= Float32(1.9999999867631625e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(2.0) + Float32(x / s))));
	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);
	elseif (t_0 <= single(1.9999999867631625e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s) < 1.99999999e37

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 52.3%

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

      1. mul-1-neg 52.3%

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

      2. unsub-neg 52.3%

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

   4. Simplified 52.3%

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

      1. sub-neg 52.3%

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

      2. neg-mul-1 52.3%

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

      3. add-log-exp 94.8%

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

      4. log-pow 94.8%

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

      5. metadata-eval 94.8%

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

      6. pow-pow 94.9%

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

      7. flip-+ 47.2%

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

   6. Applied egg-rr 71.6%

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

   if 1.99999999e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 7: 50.2% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -1.0)
   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) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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) / Float32(-s)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.5%

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

      1. distribute-frac-neg 99.5%

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

      2. exp-neg 99.5%

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

      3. add-sqr-sqrt 19.2%

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

      4. sqrt-unprod 40.1%

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

      5. sqr-neg 40.1%

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

      6. sqrt-unprod 21.2%

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

      7. add-sqr-sqrt 39.3%

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

      8. add-cbrt-cube 39.3%

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

      9. pow1/3 39.3%

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

      10. pow-flip 39.3%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in s around -inf 62.3%

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

      1. distribute-rgt-out 62.3%

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

      2. metadata-eval 62.3%

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

   6. Simplified 62.3%

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

      1. *-commutative 62.3%

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

      2. add-sqr-sqrt 18.9%

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

      3. sqrt-unprod 64.3%

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

      4. sqr-neg 64.3%

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

      5. sqrt-unprod 42.1%

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

      6. add-sqr-sqrt 60.3%

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

      7. distribute-rgt-neg-in 60.3%

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

      8. distribute-lft-neg-in 60.3%

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

      9. metadata-eval 60.3%

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

      10. associate-*l/ 60.8%

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

      11. associate-/r/ 60.3%

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

      12. frac-2neg 60.3%

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

      13. associate-/r/ 60.8%

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

      14. add-sqr-sqrt 42.7%

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

      15. sqrt-unprod 64.3%

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

      16. sqr-neg 64.3%

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

      17. sqrt-unprod 18.9%

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

      18. add-sqr-sqrt 62.9%

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

   8. Applied egg-rr 62.9%

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

4. Final simplification 50.7%

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

Alternative 8: 49.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.5%

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

      1. distribute-frac-neg 99.5%

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

      2. exp-neg 99.5%

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

      3. add-sqr-sqrt 19.2%

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

      4. sqrt-unprod 40.1%

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

      5. sqr-neg 40.1%

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

      6. sqrt-unprod 21.2%

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

      7. add-sqr-sqrt 39.3%

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

      8. add-cbrt-cube 39.3%

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

      9. pow1/3 39.3%

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

      10. pow-flip 39.3%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in s around -inf 62.3%

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

      1. distribute-rgt-out 62.3%

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

      2. metadata-eval 62.3%

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

   6. Simplified 62.3%

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

      1. associate-*r/ 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-*r* 62.3%

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

      4. metadata-eval 62.3%

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

      5. associate-/l* 62.3%

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

   8. Applied egg-rr 62.3%

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

      1. associate-/r/ 62.3%

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

   10. Simplified 62.3%

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

4. Final simplification 50.3%

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

Alternative 9: 49.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.5%

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

      1. distribute-frac-neg 99.5%

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

      2. exp-neg 99.5%

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

      3. add-sqr-sqrt 19.2%

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

      4. sqrt-unprod 40.1%

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

      5. sqr-neg 40.1%

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

      6. sqrt-unprod 21.2%

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

      7. add-sqr-sqrt 39.3%

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

      8. add-cbrt-cube 39.3%

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

      9. pow1/3 39.3%

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

      10. pow-flip 39.3%

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

   3. Applied egg-rr 97.6%

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

   4. Taylor expanded in s around -inf 62.3%

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

      1. distribute-rgt-out 62.3%

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

      2. metadata-eval 62.3%

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

   6. Simplified 62.3%

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

      1. associate-*r/ 62.3%

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

      2. *-commutative 62.3%

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

      3. associate-*r* 62.3%

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

      4. metadata-eval 62.3%

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

      5. associate-/l* 62.3%

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

   8. Applied egg-rr 62.3%

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

4. Final simplification 50.3%

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

Alternative 10: 49.8% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. unsub-neg 62.3%

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

   4. Simplified 62.3%

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

4. Final simplification 50.3%

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

Alternative 11: 48.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 0.05000000074505806) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		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 <= 0.05000000074505806e0) 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(0.05000000074505806))
		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(0.05000000074505806))
		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) < 0.0500000007

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 55.1%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 40.8%

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

      1. mul-1-neg 40.8%

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

      2. unsub-neg 40.8%

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

   4. Simplified 40.8%

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

   5. Taylor expanded in x around inf 40.8%

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

      1. mul-1-neg 40.8%

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

      2. distribute-frac-neg 40.8%

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

   7. Simplified 40.8%

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

4. Final simplification 49.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.05000000074505806:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 12: 46.7% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-6) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-6f) {
		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 <= (-9.999999747378752e-6)) 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(-9.999999747378752e-6))
		tmp = Float32(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(-9.999999747378752e-6))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-6

   1. Initial program 99.9%

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

   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 49.8%

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

      1. associate-*r/ 49.8%

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

      2. neg-mul-1 49.8%

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

   7. Simplified 49.8%

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

   if -9.99999975e-6 < x

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 47.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 13: 46.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.014999999664723873) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.014999999664723873f) {
		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 <= (-0.014999999664723873e0)) 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(-0.014999999664723873))
		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(-0.014999999664723873))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.0149999997

   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 56.5%

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

      1. distribute-rgt-out 56.5%

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

      2. metadata-eval 56.5%

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

   6. Simplified 56.5%

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

      1. *-commutative 56.5%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 65.1%

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

      4. sqr-neg 65.1%

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

      5. sqrt-unprod 56.5%

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

      6. add-sqr-sqrt 56.5%

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

      7. distribute-rgt-neg-in 56.5%

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

      8. distribute-lft-neg-in 56.5%

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

      9. metadata-eval 56.5%

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

      10. associate-*l/ 56.5%

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

   8. Applied egg-rr 56.5%

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

   9. Taylor expanded in s around 0 52.1%

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

   if -0.0149999997 < x

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 46.4%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.014999999664723873:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14: 35.4% 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 99.7%

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

2. Taylor expanded in x around 0 35.6%

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

3. Final simplification 35.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023315 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))