Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 15

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (-x / s)));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

3. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. exp-1-e 99.9%

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

5. Applied egg-rr 99.9%

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

   1. pow-flip 99.9%

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

   2. distribute-neg-frac 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 90.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{\frac{x}{s \cdot s}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 0.20000000298023224)
       (+ 0.5 (* (/ x s) 0.25))
       (* 2.0 (/ (/ 1.0 (/ x (* s s))) x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((1.0f / (x / (s * s))) / 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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 0.20000000298023224e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((1.0e0 / (x / (s * s))) / 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(1.0) / Float32(x / Float32(s * s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * ((single(1.0) / (x / (s * s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. unpow2 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 69.8%

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

      3. unpow2 69.8%

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

      4. associate-*l/ 70.4%

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

      5. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. clear-num 81.2%

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

      2. inv-pow 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. unpow-1 81.2%

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

       2. associate-/l/ 86.2%

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

   11. Simplified 86.2%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{\frac{x}{s \cdot s}}}{x}\\ \end{array} \]

Alternative 4: 85.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 0.20000000298023224)
       (+ 0.5 (* (/ x s) 0.25))
       (* 2.0 (* (/ s x) (/ s x)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 0.20000000298023224f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((s / x) * (s / 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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 0.20000000298023224e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((s / x) * (s / 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(0.20000000298023224))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * ((s / x) * (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. unpow2 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 69.8%

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

      3. unpow2 69.8%

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

      4. associate-*l/ 70.4%

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

      5. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. associate-/l/ 70.4%

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

      2. clear-num 70.4%

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

      3. div-inv 70.4%

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

      4. associate-/r/ 69.9%

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

   9. Applied egg-rr 69.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \end{array} \]

Alternative 5: 85.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 0.20000000298023224)
       (+ 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 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 0.20000000298023224f) {
		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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 0.20000000298023224e0) 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(0.20000000298023224))
		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(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(0.20000000298023224))
		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) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. unpow2 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 69.8%

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

      3. unpow2 69.8%

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

      4. associate-*l/ 70.4%

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

      5. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. associate-/l/ 70.4%

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

      2. clear-num 70.4%

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

      3. div-inv 70.4%

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

      4. div-inv 70.4%

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

      5. div-inv 70.4%

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

      6. clear-num 70.4%

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

   9. Applied egg-rr 70.4%

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

       1. associate-*r/ 70.4%

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

       2. *-rgt-identity 70.4%

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

   11. Simplified 70.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 6: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 100000:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 100000.0)
       (+ 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 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 100000.0f) {
		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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 100000.0e0) 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(100000.0))
		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(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(100000.0))
		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) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 1e5

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 87.7%

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

      1. *-commutative 87.7%

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

   4. Simplified 87.7%

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

   if 1e5 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. unpow2 86.3%

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

      4. unpow2 86.3%

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

      5. times-frac 77.1%

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

   4. Simplified 77.1%

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

   5. Taylor expanded in x around inf 84.9%

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

      1. unpow2 84.9%

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

      2. associate-/l* 74.2%

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

      3. unpow2 74.2%

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

      4. associate-*l/ 74.5%

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

      5. associate-/r* 80.2%

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

   7. Simplified 80.2%

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

   8. Taylor expanded in s around 0 84.9%

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

      1. unpow2 84.9%

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

      2. unpow2 84.9%

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

   10. Simplified 84.9%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 100000:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 7: 89.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 0.20000000298023224)
       (+ 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 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 0.20000000298023224f) {
		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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 0.20000000298023224e0) 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(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(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(0.20000000298023224))
		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) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. unpow2 80.9%

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

      4. unpow2 80.9%

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

      5. times-frac 72.8%

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

   4. Simplified 72.8%

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

   5. Taylor expanded in x around inf 79.6%

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

      1. unpow2 79.6%

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

      2. associate-/l* 69.8%

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

      3. unpow2 69.8%

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

      4. associate-*l/ 70.4%

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

      5. associate-/r* 75.7%

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

   7. Simplified 75.7%

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

      1. associate-*r/ 75.7%

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

      2. div-inv 75.7%

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

      3. clear-num 75.7%

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

   9. Applied egg-rr 75.7%

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

       1. associate-*r/ 86.0%

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

   11. Applied egg-rr 86.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \]

Alternative 8: 74.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (- 1.0 (/ s x))
     (if (<= t_0 0.20000000298023224) (+ 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 <= -10.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 0.20000000298023224f) {
		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 <= (-10.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 0.20000000298023224e0) 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(-10.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(0.20000000298023224))
		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(-10.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(0.20000000298023224))
		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) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

      1. +-commutative 95.6%

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

      2. mul-1-neg 95.6%

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

      3. unsub-neg 95.6%

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

   7. Simplified 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. unsub-neg 41.7%

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

   4. Simplified 41.7%

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

   5. Taylor expanded in x around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

   7. Simplified 41.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 9: 74.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 0.20000000298023224) (+ 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 <= -10.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 0.20000000298023224f) {
		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 <= (-10.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 0.20000000298023224e0) 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(-10.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(0.20000000298023224))
		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(-10.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(0.20000000298023224))
		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) < -10

   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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 95.6%

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

   5. Taylor expanded in x around inf 95.6%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 96.3%

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

      1. *-commutative 96.3%

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

   4. Simplified 96.3%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. unsub-neg 41.7%

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

   4. Simplified 41.7%

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

   5. Taylor expanded in x around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

   7. Simplified 41.7%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 10: 72.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (- 1.0 (/ s x))
     (if (<= t_0 0.20000000298023224) 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 - (s / x);
	} else if (t_0 <= 0.20000000298023224f) {
		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 - (s / x)
    else if (t_0 <= 0.20000000298023224e0) 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(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(0.20000000298023224))
		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) - (s / x);
	elseif (t_0 <= single(0.20000000298023224))
		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 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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   if -2 < (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 90.1%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 41.7%

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

      1. mul-1-neg 41.7%

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

      2. unsub-neg 41.7%

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

   4. Simplified 41.7%

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

   5. Taylor expanded in x around inf 41.7%

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

      1. mul-1-neg 41.7%

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

      2. distribute-frac-neg 41.7%

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

   7. Simplified 41.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 11: 89.5% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 4.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 2.0f * ((1.0f / (x / (s * 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) <= 4.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 2.0e0 * ((1.0e0 / (x / (s * s))) / x)
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 4

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 94.6%

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

   if 4 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 81.6%

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

      1. mul-1-neg 81.6%

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

      2. unsub-neg 81.6%

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

      3. unpow2 81.6%

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

      4. unpow2 81.6%

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

      5. times-frac 73.2%

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

   4. Simplified 73.2%

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

   5. Taylor expanded in x around inf 80.4%

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

      1. unpow2 80.4%

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

      2. associate-/l* 70.4%

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

      3. unpow2 70.4%

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

      4. associate-*l/ 70.8%

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

      5. associate-/r* 76.2%

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

   7. Simplified 76.2%

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

      1. clear-num 81.7%

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

      2. inv-pow 81.7%

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

   9. Applied egg-rr 81.7%

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

       1. unpow-1 81.7%

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

       2. associate-/l/ 87.0%

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

   11. Simplified 87.0%

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

4. Final simplification 91.4%

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

Alternative 12: 74.3% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} 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 = 1.0e0 / (1.0e0 + (s / x))
    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(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	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(1.0) / (single(1.0) + (s / x));
	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. 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. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.1%

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

   5. Taylor expanded in x around inf 94.1%

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

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

   1. Initial program 99.8%

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

   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 72.8%

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

Alternative 13: 67.7% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026)
   (/ (- s) x)
   (if (<= x 2.0000000072549875e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		tmp = -s / x;
	} else if (x <= 2.0000000072549875e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-0.0020000000949949026e0)) then
        tmp = -s / x
    else if (x <= 2.0000000072549875e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0020000000949949026))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(2.0000000072549875e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0020000000949949026))
		tmp = -s / x;
	elseif (x <= single(2.0000000072549875e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00200000009

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. unsub-neg 53.7%

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

   4. Simplified 53.7%

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

   5. Taylor expanded in x around inf 48.6%

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

      1. associate-*r/ 48.6%

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

      2. neg-mul-1 48.6%

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

   7. Simplified 48.6%

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

   if -0.00200000009 < x < 2.00000001e-15

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 59.7%

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

   if 2.00000001e-15 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 96.1%

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

   5. Taylor expanded in x around inf 88.9%

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

      1. +-commutative 88.9%

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

      2. mul-1-neg 88.9%

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

      3. unsub-neg 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.0000000072549875 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 14: 46.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0020000000949949026) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0020000000949949026f) {
		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 <= (-0.0020000000949949026e0)) 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(-0.0020000000949949026))
		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(-0.0020000000949949026))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00200000009

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. unsub-neg 53.7%

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

   4. Simplified 53.7%

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

   5. Taylor expanded in x around inf 48.6%

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

      1. associate-*r/ 48.6%

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

      2. neg-mul-1 48.6%

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

   7. Simplified 48.6%

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

   if -0.00200000009 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0020000000949949026:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 15: 34.8% 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 35.1%

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

3. Final simplification 35.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))