Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.3s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% 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 99.5%

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

   1. div-inv 99.6%

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

   2. exp-prod 81.2%

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

   3. neg-mul-1 81.2%

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

   4. exp-prod 81.2%

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

   5. pow-pow 99.6%

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

   6. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

   1. add-exp-log 99.5%

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

   2. clear-num 99.5%

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

   3. log-rec 99.5%

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

   4. pow-exp 99.5%

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

   5. add-log-exp 99.5%

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

   6. log-pow 99.5%

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

   7. metadata-eval 99.5%

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

   8. sqrt-pow2 99.5%

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

   9. add-exp-log 99.5%

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

   10. log-rec 99.6%

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

   11. clear-num 99.6%

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

   12. log-rec 99.5%

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

   13. log1p-udef 99.6%

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

5. Applied egg-rr 99.6%

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

   1. distribute-neg-frac 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

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 99.5%

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

   1. div-inv 99.6%

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

   2. exp-prod 81.2%

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

   3. neg-mul-1 81.2%

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

   4. exp-prod 81.2%

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

   5. pow-pow 99.6%

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

   6. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. div-inv 99.6%

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

   2. exp-prod 81.2%

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

   3. neg-mul-1 81.2%

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

   4. exp-prod 81.2%

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

   5. pow-pow 99.6%

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

   6. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

   1. inv-pow 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

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 99.5%

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

2. Final simplification 99.5%

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

Alternative 5: 62.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.00019999999494757503:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{1}{\frac{\frac{s \cdot s}{x}}{x}} - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.00019999999494757503)
   0.5
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ 1.0 (/ (/ (* s s) x) x))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.00019999999494757503f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (1.0f / (((s * s) / x) / x))) - (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) <= 0.00019999999494757503e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (1.0e0 / (((s * s) / x) / x))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.00019999999494757503))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(1.0) / Float32(Float32(Float32(s * s) / x) / x))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.00019999999494757503))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (single(1.0) / (((s * s) / x) / x))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1.99999995e-4

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 50.2%

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

   if 1.99999995e-4 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 72.8%

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

      1. mul-1-neg 72.8%

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

      2. unsub-neg 72.8%

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

      3. unpow2 72.8%

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

      4. unpow2 72.8%

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

      5. times-frac 69.5%

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

   4. Simplified 69.5%

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

      1. clear-num 69.5%

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

      2. clear-num 69.5%

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

      3. frac-times 69.5%

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

      4. metadata-eval 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. associate-*r/ 74.1%

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

      2. associate-*l/ 79.5%

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

   8. Simplified 79.5%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{1}{\frac{\frac{s \cdot s}{x}}{x}}} - \frac{x}{s}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.00019999999494757503:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{1}{\frac{\frac{s \cdot s}{x}}{x}} - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 6: 60.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * ((x / s) * (x / s))) - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((x / s) * (x / s))) - (x / s)))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. unpow2 74.1%

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

      4. unpow2 74.1%

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

      5. times-frac 80.1%

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

   4. Simplified 80.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.8%

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

Alternative 7: 61.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{s \cdot \frac{s}{x}} - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. unsub-neg 74.1%

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

      3. unpow2 74.1%

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

      4. unpow2 74.1%

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

      5. times-frac 80.1%

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

   4. Simplified 80.1%

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

      1. clear-num 80.1%

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

      2. frac-times 83.0%

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

      3. *-un-lft-identity 83.0%

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

   6. Applied egg-rr 83.0%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} - \frac{x}{s}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

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

Alternative 8: 61.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 5.000000156871975e-23)
   0.5
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 5.000000156871975e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (0.5f * ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 5.000000156871975e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(5.000000156871975e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(5.000000156871975e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (single(0.5) * ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 5.00000016e-23

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 45.6%

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

   if 5.00000016e-23 < (neg.f32 x)

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 83.1%

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

      1. mul-1-neg 83.1%

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

      2. unsub-neg 83.1%

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

      3. unpow2 83.1%

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

      4. unpow2 83.1%

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

      5. times-frac 77.5%

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

   4. Simplified 77.5%

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

      1. frac-times 83.1%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in x around inf 81.9%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot \frac{{x}^{2}}{{s}^{2}}}} \]
   8. Step-by-step derivation

      1. *-commutative 81.9%

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{{x}^{2}}{{s}^{2}} \cdot 0.5}} \]

      2. unpow2 81.9%

         \[\leadsto \frac{1}{2 + \frac{\color{blue}{x \cdot x}}{{s}^{2}} \cdot 0.5} \]

      3. unpow2 81.9%

         \[\leadsto \frac{1}{2 + \frac{x \cdot x}{\color{blue}{s \cdot s}} \cdot 0.5} \]

   9. Simplified 81.9%

      \[\leadsto \frac{1}{2 + \color{blue}{\frac{x \cdot x}{s \cdot s} \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 9: 57.5% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 49.9%

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

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

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 74.2%

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

      1. mul-1-neg 74.2%

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

      2. unsub-neg 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 69.4%

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

   4. Simplified 69.4%

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

   5. Taylor expanded in x around inf 72.0%

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

      1. unpow2 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 55.5%

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

Alternative 10: 60.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 100000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 100000.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 100000.0f) {
		tmp = 0.5f;
	} 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) :: tmp
    if ((-x / s) <= 100000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(100000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(100000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 47.5%

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

   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 83.5%

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

      1. mul-1-neg 83.5%

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

      2. unsub-neg 83.5%

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

      3. unpow2 83.5%

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

      4. unpow2 83.5%

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

      5. times-frac 77.4%

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

   4. Simplified 77.4%

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

   5. Taylor expanded in x around inf 81.1%

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

      1. unpow2 81.1%

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

      2. unpow2 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 57.7%

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

Alternative 11: 48.3% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 64.4%

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

      1. mul-1-neg 64.4%

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

      2. unsub-neg 64.4%

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

   4. Simplified 64.4%

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

4. Final simplification 49.5%

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

Alternative 12: 47.0% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 49.9%

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

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

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 44.3%

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

      1. mul-1-neg 44.3%

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

      2. unsub-neg 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in x around inf 44.3%

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

      1. neg-mul-1 44.3%

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

      2. distribute-frac-neg 44.3%

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

   7. Simplified 44.3%

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

4. Final simplification 47.9%

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

Alternative 13: 45.4% 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 99.8%

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

   2. Taylor expanded in x around 0 58.5%

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

      1. mul-1-neg 58.5%

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

      2. unsub-neg 58.5%

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

   4. Simplified 58.5%

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

   5. Taylor expanded in x around inf 52.5%

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

      1. associate-*r/ 52.5%

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

      2. mul-1-neg 52.5%

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

   7. Simplified 52.5%

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

   if -0.00200000009 < x

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 44.3%

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

4. Final simplification 46.3%

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

Alternative 14: 34.2% 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 99.5%

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

2. Taylor expanded in x around 0 34.9%

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

3. Final simplification 34.9%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023188 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))