Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.0s

Alternatives: 20

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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (sqrt(exp(single(-1.0))) ^ ((x / s) * single(2.0))));
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. add-exp-log 99.8%

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

   2. inv-pow 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

   5. pow-exp 99.8%

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

   6. add-sqr-sqrt 99.8%

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

   7. unpow-prod-down 99.8%

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

5. Applied egg-rr 99.8%

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

   1. pow-sqr 99.8%

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

   2. count-2 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. distribute-lft-out 99.8%

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

   6. metadata-eval 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-2.0)) ^ ((x / s) * single(0.5))));
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. add-exp-log 99.8%

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

   2. inv-pow 99.8%

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

   3. log-pow 99.8%

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

   4. add-log-exp 99.8%

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

   5. pow-exp 99.8%

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

   6. sqr-pow 99.8%

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

   7. pow-prod-down 99.7%

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

   8. prod-exp 99.8%

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

   9. metadata-eval 99.8%

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

   10. div-inv 99.8%

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

   11. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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. Final simplification 99.8%

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

Alternative 4: 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 5: 88.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{s \cdot s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -8.999999840607448e-23)
   (/ 1.0 (+ 2.0 (- (* 0.5 (* (* x x) (/ 1.0 (* s s)))) (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -8.999999840607448e-23f) {
		tmp = 1.0f / (2.0f + ((0.5f * ((x * x) * (1.0f / (s * s)))) - (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-8.999999840607448e-23)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((x * x) * (1.0e0 / (s * s)))) - (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-8.999999840607448e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x * x) * Float32(Float32(1.0) / Float32(s * s)))) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-8.999999840607448e-23))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * ((x * x) * (single(1.0) / (s * s)))) - (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -8.99999984e-23

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. frac-times 80.5%

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

   6. Applied egg-rr 80.5%

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

      1. div-inv 81.1%

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

   8. Applied egg-rr 81.1%

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

   if -8.99999984e-23 < 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 90.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{s \cdot s}\right) - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 6: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -8.999999840607448e-23)
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ (* x x) (* s s))) (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -8.999999840607448e-23f) {
		tmp = 1.0f / (2.0f + ((0.5f * ((x * x) / (s * s))) - (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-8.999999840607448e-23)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((x * x) / (s * s))) - (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-8.999999840607448e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s))) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-8.999999840607448e-23))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * ((x * x) / (s * s))) - (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -8.99999984e-23

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. frac-times 80.5%

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

   6. Applied egg-rr 80.5%

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

   if -8.99999984e-23 < 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 90.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 7: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{0.5}{\frac{s \cdot s}{x \cdot x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -8.999999840607448e-23)
   (/ 1.0 (+ 2.0 (- (/ 0.5 (/ (* s s) (* x x))) (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -8.999999840607448e-23f) {
		tmp = 1.0f / (2.0f + ((0.5f / ((s * s) / (x * x))) - (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-8.999999840607448e-23)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 / ((s * s) / (x * x))) - (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-8.999999840607448e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) / Float32(Float32(s * s) / Float32(x * x))) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-8.999999840607448e-23))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) / ((s * s) / (x * x))) - (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -8.99999984e-23

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.6%

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

   4. Simplified 75.6%

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

      1. associate-*r* 75.0%

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

      2. clear-num 75.0%

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

      3. un-div-inv 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. associate-/l* 75.0%

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

      2. associate-/r* 75.0%

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

      3. associate-*r/ 75.0%

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

      4. unpow2 75.0%

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

      5. associate-/l* 80.5%

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

      6. unpow2 80.5%

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

   8. Simplified 80.5%

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

   if -8.99999984e-23 < 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 90.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{0.5}{\frac{s \cdot s}{x \cdot x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 8: 88.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) - \frac{\left(x \cdot x\right) \cdot -0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -8.999999840607448e-23)
   (/ 1.0 (- (- 2.0 (/ x s)) (/ (* (* x x) -0.5) (* s s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -8.999999840607448e-23f) {
		tmp = 1.0f / ((2.0f - (x / s)) - (((x * x) * -0.5f) / (s * s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-8.999999840607448e-23)) then
        tmp = 1.0e0 / ((2.0e0 - (x / s)) - (((x * x) * (-0.5e0)) / (s * s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-8.999999840607448e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(x / s)) - Float32(Float32(Float32(x * x) * Float32(-0.5)) / Float32(s * s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-8.999999840607448e-23))
		tmp = single(1.0) / ((single(2.0) - (x / s)) - (((x * x) * single(-0.5)) / (s * s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -8.99999984e-23

   1. Initial program 99.6%

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

      1. distribute-frac-neg 99.6%

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

      2. exp-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in s around inf 45.3%

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

      1. +-commutative 45.3%

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

      2. mul-1-neg 45.3%

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

      3. sub-neg 45.3%

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

      4. mul-1-neg 45.3%

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

      5. unsub-neg 45.3%

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

      6. distribute-rgt-out 80.5%

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

      7. metadata-eval 80.5%

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

      8. unpow2 80.5%

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

      9. times-frac 75.0%

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

      10. unpow2 75.0%

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

   6. Simplified 75.0%

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

      1. frac-times 80.5%

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

   8. Applied egg-rr 80.5%

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

   if -8.99999984e-23 < 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 90.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) - \frac{\left(x \cdot x\right) \cdot -0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 9: 87.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -8.999999840607448e-23)
   (/ 1.0 (+ 2.0 (- (/ (* x x) (* s s)) (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -8.999999840607448e-23f) {
		tmp = 1.0f / (2.0f + (((x * x) / (s * s)) - (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-8.999999840607448e-23)) then
        tmp = 1.0e0 / (2.0e0 + (((x * x) / (s * s)) - (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-8.999999840607448e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * x) / Float32(s * s)) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-8.999999840607448e-23))
		tmp = single(1.0) / (single(2.0) + (((x * x) / (s * s)) - (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -8.99999984e-23

   1. Initial program 99.6%

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

      1. distribute-frac-neg 99.6%

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

      2. exp-neg 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in x around 0 15.7%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. unpow2 80.4%

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

      2. unpow2 80.4%

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

      3. neg-mul-1 80.4%

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

      4. distribute-neg-frac 80.4%

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

   7. Simplified 80.4%

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

   if -8.99999984e-23 < 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 90.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.999999840607448 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 10: 84.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.0000000843119176e-17)
   (/ (/ (* 2.0 (* s s)) x) x)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.0000000843119176e-17f) {
		tmp = ((2.0f * (s * s)) / x) / x;
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (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 <= (-5.0000000843119176e-17)) then
        tmp = ((2.0e0 * (s * s)) / x) / x
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(Float32(2.0) * Float32(s * s)) / x) / x);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.0000000843119176e-17))
		tmp = ((single(2.0) * (s * s)) / x) / x;
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000008e-17

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x around inf 78.3%

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

      1. unpow2 78.3%

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

      2. unpow2 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in s around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. associate-/r* 78.3%

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

   10. Simplified 78.3%

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

   if -5.00000008e-17 < x

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.8%

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

   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 89.1%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 11: 66.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14)
   (* s (/ -1.0 x))
   (if (<= x 1.600000050199032e-21) (+ 0.5 (* (/ x s) 0.25)) (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		tmp = s * (-1.0f / x);
	} else if (x <= 1.600000050199032e-21f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} 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 <= (-3.99999992980668e-14)) then
        tmp = s * ((-1.0e0) / x)
    else if (x <= 1.600000050199032e-21) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.99999992980668e-14))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	elseif (x <= Float32(1.600000050199032e-21))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	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(-3.99999992980668e-14))
		tmp = s * (single(-1.0) / x);
	elseif (x <= single(1.600000050199032e-21))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. unsub-neg 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   7. Simplified 40.3%

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

      1. div-inv 40.3%

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

   9. Applied egg-rr 40.3%

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

   if -3.99999993e-14 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 71.2%

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

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. neg-mul-1 88.4%

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

      2. distribute-frac-neg 88.4%

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

      3. +-commutative 88.4%

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

      4. distribute-frac-neg 88.4%

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

      5. unsub-neg 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 12: 67.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14)
   (* s (/ -1.0 x))
   (if (<= x 1.600000050199032e-21)
     (+ 0.5 (* (/ x s) 0.25))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		tmp = s * (-1.0f / x);
	} else if (x <= 1.600000050199032e-21f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.99999992980668e-14)) then
        tmp = s * ((-1.0e0) / x)
    else if (x <= 1.600000050199032e-21) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.99999992980668e-14))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	elseif (x <= Float32(1.600000050199032e-21))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.99999992980668e-14))
		tmp = s * (single(-1.0) / x);
	elseif (x <= single(1.600000050199032e-21))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. unsub-neg 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   7. Simplified 40.3%

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

      1. div-inv 40.3%

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

   9. Applied egg-rr 40.3%

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

   if -3.99999993e-14 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 71.2%

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

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 89.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 13: 79.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14)
   (* 2.0 (* (/ s x) (/ s x)))
   (if (<= x 1.600000050199032e-21)
     (+ 0.5 (* (/ x s) 0.25))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		tmp = 2.0f * ((s / x) * (s / x));
	} else if (x <= 1.600000050199032e-21f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.99999992980668e-14)) then
        tmp = 2.0e0 * ((s / x) * (s / x))
    else if (x <= 1.600000050199032e-21) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.99999992980668e-14))
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	elseif (x <= Float32(1.600000050199032e-21))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.99999992980668e-14))
		tmp = single(2.0) * ((s / x) * (s / x));
	elseif (x <= single(1.600000050199032e-21))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 79.7%

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

      1. mul-1-neg 79.7%

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

      2. unsub-neg 79.7%

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

      3. unpow2 79.7%

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

      4. unpow2 79.7%

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

      5. times-frac 77.8%

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

   4. Simplified 77.8%

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

   5. Taylor expanded in x around inf 78.7%

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

      1. unpow2 78.7%

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

      2. unpow2 78.7%

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

      3. times-frac 76.5%

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

   7. Simplified 76.5%

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

   if -3.99999993e-14 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 71.2%

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

      1. *-commutative 71.2%

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

   4. Simplified 71.2%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 89.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 14: 80.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.0000000843119176e-17)
   (* 2.0 (/ (* s s) (* x x)))
   (if (<= x 1.600000050199032e-21)
     (+ 0.5 (* (/ x s) 0.25))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.0000000843119176e-17f) {
		tmp = 2.0f * ((s * s) / (x * x));
	} else if (x <= 1.600000050199032e-21f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.0000000843119176e-17)) then
        tmp = 2.0e0 * ((s * s) / (x * x))
    else if (x <= 1.600000050199032e-21) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0000000843119176e-17))
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	elseif (x <= Float32(1.600000050199032e-21))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.0000000843119176e-17))
		tmp = single(2.0) * ((s * s) / (x * x));
	elseif (x <= single(1.600000050199032e-21))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000008e-17

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x around inf 78.3%

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

      1. unpow2 78.3%

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

      2. unpow2 78.3%

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

   7. Simplified 78.3%

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

   if -5.00000008e-17 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. *-commutative 74.8%

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

   4. Simplified 74.8%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 89.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 15: 80.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.0000000843119176e-17)
   (/ (/ (* 2.0 (* s s)) x) x)
   (if (<= x 1.600000050199032e-21)
     (+ 0.5 (* (/ x s) 0.25))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.0000000843119176e-17f) {
		tmp = ((2.0f * (s * s)) / x) / x;
	} else if (x <= 1.600000050199032e-21f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.0000000843119176e-17)) then
        tmp = ((2.0e0 * (s * s)) / x) / x
    else if (x <= 1.600000050199032e-21) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0000000843119176e-17))
		tmp = Float32(Float32(Float32(Float32(2.0) * Float32(s * s)) / x) / x);
	elseif (x <= Float32(1.600000050199032e-21))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.0000000843119176e-17))
		tmp = ((single(2.0) * (s * s)) / x) / x;
	elseif (x <= single(1.600000050199032e-21))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.00000008e-17

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 80.5%

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

      1. mul-1-neg 80.5%

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

      2. unsub-neg 80.5%

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

      3. unpow2 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 75.9%

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

   4. Simplified 75.9%

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

   5. Taylor expanded in x around inf 78.3%

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

      1. unpow2 78.3%

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

      2. unpow2 78.3%

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

   7. Simplified 78.3%

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

   8. Taylor expanded in s around 0 78.3%

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

      1. associate-*r/ 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. associate-/r* 78.3%

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

   10. Simplified 78.3%

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

   if -5.00000008e-17 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. *-commutative 74.8%

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

   4. Simplified 74.8%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 89.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 16: 66.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14)
   (/ (- s) x)
   (if (<= x 1.600000050199032e-21) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		tmp = -s / x;
	} else if (x <= 1.600000050199032e-21f) {
		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 <= (-3.99999992980668e-14)) then
        tmp = -s / x
    else if (x <= 1.600000050199032e-21) 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(-3.99999992980668e-14))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(1.600000050199032e-21))
		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(-3.99999992980668e-14))
		tmp = -s / x;
	elseif (x <= single(1.600000050199032e-21))
		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 < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. unsub-neg 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   7. Simplified 40.3%

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

   if -3.99999993e-14 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 67.6%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. neg-mul-1 88.4%

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

      2. distribute-frac-neg 88.4%

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

      3. +-commutative 88.4%

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

      4. distribute-frac-neg 88.4%

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

      5. unsub-neg 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 17: 66.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14)
   (* s (/ -1.0 x))
   (if (<= x 1.600000050199032e-21) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		tmp = s * (-1.0f / x);
	} else if (x <= 1.600000050199032e-21f) {
		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 <= (-3.99999992980668e-14)) then
        tmp = s * ((-1.0e0) / x)
    else if (x <= 1.600000050199032e-21) 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(-3.99999992980668e-14))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	elseif (x <= Float32(1.600000050199032e-21))
		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(-3.99999992980668e-14))
		tmp = s * (single(-1.0) / x);
	elseif (x <= single(1.600000050199032e-21))
		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 < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. unsub-neg 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   7. Simplified 40.3%

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

      1. div-inv 40.3%

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

   9. Applied egg-rr 40.3%

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

   if -3.99999993e-14 < x < 1.60000005e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 67.6%

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

   if 1.60000005e-21 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 88.4%

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

      1. neg-mul-1 88.4%

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

      2. distribute-frac-neg 88.4%

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

      3. +-commutative 88.4%

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

      4. distribute-frac-neg 88.4%

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

      5. unsub-neg 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.600000050199032 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 18: 68.3% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 5.000000136226006e-28)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 5.000000136226006e-28f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 5.000000136226006e-28) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(5.000000136226006e-28))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(5.000000136226006e-28))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 5.00000014e-28

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 54.8%

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

      1. mul-1-neg 54.8%

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

      2. unsub-neg 54.8%

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

   4. Simplified 54.8%

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

   if 5.00000014e-28 < x

   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 94.5%

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

   5. Taylor expanded in x around inf 85.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]

Alternative 19: 44.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.99999992980668e-14) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.99999992980668e-14f) {
		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 <= (-3.99999992980668e-14)) 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(-3.99999992980668e-14))
		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(-3.99999992980668e-14))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.99999993e-14

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 42.4%

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

      1. mul-1-neg 42.4%

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

      2. unsub-neg 42.4%

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

   4. Simplified 42.4%

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

   5. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   7. Simplified 40.3%

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

   if -3.99999993e-14 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 47.2%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.99999992980668 \cdot 10^{-14}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 20: 34.0% 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 33.6%

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

3. Final simplification 33.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023182 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))