Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 11.2s

Alternatives: 17

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 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 84.1%

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

   3. neg-mul-1 84.1%

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

   4. exp-prod 84.1%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

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

   4. pow-exp 99.9%

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

   5. neg-mul-1 99.9%

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

   6. distribute-neg-frac 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

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 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 84.1%

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

   3. neg-mul-1 84.1%

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

   4. exp-prod 84.1%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (+ 1.0 (expm1 (/ (- x) s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (1.0f + expm1f((-x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + expm1(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 84.1%

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

   3. neg-mul-1 84.1%

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

   4. exp-prod 84.1%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. expm1-log1p-u 99.8%

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

   2. expm1-udef 99.8%

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

   3. log1p-udef 99.8%

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

   4. add-exp-log 99.8%

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

   5. pow-exp 99.8%

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

   6. neg-mul-1 99.8%

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

   7. distribute-neg-frac 99.8%

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

5. Applied egg-rr 99.8%

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

   1. associate--l+ 99.8%

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

   2. expm1-def 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + \left(1 + \mathsf{expm1}\left(\frac{-x}{s}\right)\right)} \]

Alternative 4: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 5: 91.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{0.5 \cdot \frac{x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = (1.0f / x) / (0.5f * (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 = -x / s
    if (t_0 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = (1.0e0 / x) / (0.5e0 * (x / (s * 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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(0.5) * Float32(x / Float32(s * s))));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. mul-1-neg 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. unpow2 84.1%

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

      6. times-frac 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

   7. Simplified 83.1%

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

      1. associate-/l* 71.1%

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

      2. associate-/r/ 70.8%

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

   9. Applied egg-rr 70.8%

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

       1. associate-*l/ 83.1%

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

       2. associate-*r/ 83.1%

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

       3. *-commutative 83.1%

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

       4. associate-/l/ 85.2%

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

       5. div-inv 85.2%

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

       6. clear-num 86.7%

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

       7. associate-*l/ 86.7%

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

       8. *-un-lft-identity 86.7%

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

       9. *-un-lft-identity 86.7%

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

       10. *-commutative 86.7%

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

       11. times-frac 86.7%

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

       12. metadata-eval 86.7%

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

   11. Applied egg-rr 86.7%

       \[\leadsto \color{blue}{\frac{\frac{1}{x}}{0.5 \cdot \frac{x}{s \cdot s}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{0.5 \cdot \frac{x}{s \cdot s}}\\ \end{array} \]

Alternative 6: 86.7% accurate, 5.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * (s * (s / (x * x)));
	}
	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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * (s * (s / (x * x)))
    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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. mul-1-neg 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. unpow2 84.1%

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

      6. times-frac 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

   7. Simplified 83.1%

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

      1. associate-/l* 71.1%

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

      2. associate-/r/ 70.8%

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

   9. Applied egg-rr 70.8%

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

4. Final simplification 87.8%

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

Alternative 7: 86.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * (s / (x * (x / 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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * (s / (x * (x / 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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * Float32(x / s))));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. mul-1-neg 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. unpow2 84.1%

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

      6. times-frac 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

      3. times-frac 70.9%

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

      4. unpow2 70.9%

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

   7. Simplified 70.9%

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

      1. unpow2 70.9%

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

      2. clear-num 70.9%

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

      3. frac-times 71.2%

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

      4. *-un-lft-identity 71.2%

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

   9. Applied egg-rr 71.2%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 8: 89.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. mul-1-neg 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. unpow2 84.1%

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

      6. times-frac 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

   7. Simplified 83.1%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 9: 91.3% accurate, 5.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = ((2.0f * (s * s)) / x) / x;
	}
	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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = ((2.0e0 * (s * s)) / x) / x
    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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(Float32(2.0) * Float32(s * s)) / x) / x);
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

      2. mul-1-neg 84.1%

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

      3. unsub-neg 84.1%

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

      4. unpow2 84.1%

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

      5. unpow2 84.1%

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

      6. times-frac 72.5%

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

   4. Simplified 72.5%

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

   5. Taylor expanded in x around inf 83.1%

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

      1. unpow2 83.1%

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

      2. unpow2 83.1%

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

   7. Simplified 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-/r* 85.2%

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

      3. *-commutative 85.2%

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

   9. Applied egg-rr 85.2%

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

4. Final simplification 93.1%

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

Alternative 10: 76.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} 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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	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(-2.0))
		tmp = single(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	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) < -2

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 96.0%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.0%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 11: 74.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0) 1.0 (if (<= t_0 2.0) 0.5 (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 2.0f) {
		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 <= (-2.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 2.0e0) 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(-2.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(2.0))
		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(-2.0))
		tmp = single(1.0);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5);
	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) < -2

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 85.1%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   4. Simplified 40.9%

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

   5. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   7. Simplified 40.9%

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

4. Final simplification 74.2%

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

Alternative 12: 75.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f;
	} 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) <= (-2.0e0)) then
        tmp = 1.0e0
    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(-2.0))
		tmp = Float32(1.0);
	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(-2.0))
		tmp = single(1.0);
	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) < -2

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 85.6%

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

      3. neg-mul-1 85.6%

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

      4. exp-prod 85.6%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 2.7%

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

      1. *-inverses 98.7%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 98.7%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 61.4%

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

      1. mul-1-neg 61.4%

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

      2. unsub-neg 61.4%

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

   4. Simplified 61.4%

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

4. Final simplification 76.0%

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

Alternative 13: 70.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.7699999688147727e-8)
   (/ 1.0 (/ x s))
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.7699999688147727e-8f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.7699999688147727e-8)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.7699999688147727e-8))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.7699999688147727e-8))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.76999997e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. unsub-neg 47.9%

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

   4. Simplified 47.9%

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

   5. Taylor expanded in x around inf 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

   7. Simplified 44.8%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 62.8%

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

      3. sqr-neg 62.8%

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

      4. sqrt-prod 44.7%

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

      5. add-sqr-sqrt 44.7%

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

      6. clear-num 47.8%

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

      7. inv-pow 47.8%

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

   9. Applied egg-rr 47.8%

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

       1. unpow-1 47.8%

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

   11. Simplified 47.8%

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

   if -1.76999997e-8 < x < 4.9999998e-22

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 65.5%

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

   if 4.9999998e-22 < x

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 87.5%

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

      3. neg-mul-1 87.5%

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

      4. exp-prod 87.5%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 92.6%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 69.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.7699999688147727e-8)
   (/ (- s) x)
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.7699999688147727e-8f) {
		tmp = -s / x;
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.7699999688147727e-8)) then
        tmp = -s / x
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.7699999688147727e-8))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.7699999688147727e-8))
		tmp = -s / x;
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.76999997e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. unsub-neg 47.9%

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

   4. Simplified 47.9%

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

   5. Taylor expanded in x around inf 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

   7. Simplified 44.8%

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

   if -1.76999997e-8 < x < 4.9999998e-22

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 65.5%

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

   if 4.9999998e-22 < x

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 87.5%

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

      3. neg-mul-1 87.5%

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

      4. exp-prod 87.5%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 92.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 68.9% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.7699999688147727e-8)
   (/ s x)
   (if (<= x 4.999999841327613e-22) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.7699999688147727e-8f) {
		tmp = s / x;
	} else if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.7699999688147727e-8)) then
        tmp = s / x
    else if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.7699999688147727e-8))
		tmp = Float32(s / x);
	elseif (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.7699999688147727e-8))
		tmp = s / x;
	elseif (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.76999997e-8

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 47.9%

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

      1. mul-1-neg 47.9%

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

      2. unsub-neg 47.9%

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

   4. Simplified 47.9%

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

   5. Taylor expanded in x around inf 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

   7. Simplified 44.8%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 62.8%

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

      3. sqr-neg 62.8%

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

      4. sqrt-prod 44.7%

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

      5. add-sqr-sqrt 44.7%

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

      6. expm1-log1p-u 44.7%

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

      7. expm1-udef 96.3%

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

   9. Applied egg-rr 96.3%

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

       1. expm1-def 44.7%

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

       2. expm1-log1p 44.7%

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

   11. Simplified 44.7%

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

   if -1.76999997e-8 < x < 4.9999998e-22

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 65.5%

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

   if 4.9999998e-22 < x

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 87.5%

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

      3. neg-mul-1 87.5%

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

      4. exp-prod 87.5%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 92.6%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7699999688147727 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 57.0% accurate, 34.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 4.999999841327613e-22) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999841327613e-22f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999841327613e-22) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999841327613e-22))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999841327613e-22))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.9999998e-22

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 35.1%

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

   if 4.9999998e-22 < x

   1. Initial program 99.9%

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

      1. div-inv 99.9%

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

      2. exp-prod 87.5%

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

      3. neg-mul-1 87.5%

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

      4. exp-prod 87.5%

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

      5. pow-pow 99.9%

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

      6. div-inv 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-rec 100.0%

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

      3. log1p-udef 100.0%

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

      4. pow-exp 100.0%

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

      5. neg-mul-1 100.0%

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

      6. 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. Step-by-step derivation

      1. exp-neg 99.9%

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

      2. log1p-udef 99.9%

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

      3. add-exp-log 99.9%

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

      4. add-sqr-sqrt 99.8%

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

      5. associate-/r* 99.9%

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

   7. Applied egg-rr 4.8%

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

      1. *-inverses 92.6%

         \[\leadsto \color{blue}{1} \]

   9. Simplified 92.6%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17: 33.6% 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 99.8%

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

2. Taylor expanded in x around 0 34.0%

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

3. Final simplification 34.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023278 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))