Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.2s

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.8%

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

   1. 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%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.8%

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

   4. rec-exp 99.8%

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

5. Applied egg-rr 99.8%

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

   1. distribute-neg-frac 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{3}\right)}^{\left(\frac{x \cdot -0.3333333333333333}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp 3.0) (/ (* x -0.3333333333333333) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(3.0f), ((x * -0.3333333333333333f) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp(3.0e0) ** ((x * (-0.3333333333333333e0)) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(3.0)) ^ Float32(Float32(x * Float32(-0.3333333333333333)) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(3.0)) ^ ((x * single(-0.3333333333333333)) / 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. div-inv 99.7%

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

   4. add-sqr-sqrt 51.9%

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

   5. sqrt-unprod 63.4%

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

   6. sqr-neg 63.4%

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

   7. sqrt-unprod 13.0%

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

   8. add-sqr-sqrt 27.0%

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

   9. div-inv 27.0%

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

   10. pow1 27.0%

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

   11. pow1 27.0%

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

   12. add-cbrt-cube 27.0%

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

   13. pow1/3 27.0%

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

   14. pow-flip 27.0%

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

3. Applied egg-rr 99.2%

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

   1. *-un-lft-identity 99.2%

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

   2. exp-prod 99.2%

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

   3. pow-pow 99.8%

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

5. Applied egg-rr 99.8%

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

   1. *-lft-identity 99.8%

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

   2. associate-*l/ 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 4: 91.0% accurate, 4.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 20.0f) {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x * (x * (1.0f / (s * 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 = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x * (x * (1.0e0 / (s * 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(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s))) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(1.0) / Float32(s * 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(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x * (x * (single(1.0) / (s * 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 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. Taylor expanded in x around 0 94.2%

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

      1. expm1-log1p-u 94.2%

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

      2. expm1-udef 94.2%

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

      3. log1p-udef 94.2%

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

      4. add-exp-log 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. associate--l+ 94.2%

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

      2. sub-neg 94.2%

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

      3. +-commutative 94.2%

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

      4. metadata-eval 94.2%

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

   8. Simplified 94.2%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 71.6%

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

      1. mul-1-neg 71.6%

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

      2. unsub-neg 71.6%

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

      3. unpow2 71.6%

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

      4. unpow2 71.6%

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

      5. times-frac 65.5%

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

   4. Simplified 65.5%

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

      1. clear-num 65.5%

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

      2. frac-times 67.4%

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

      3. *-un-lft-identity 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. frac-2neg 67.4%

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

      2. div-inv 73.3%

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

      3. add-sqr-sqrt 73.3%

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

      4. sqrt-unprod 67.8%

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

      5. sqr-neg 67.8%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 45.8%

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

      8. associate-*l/ 50.9%

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

      9. distribute-neg-frac 50.9%

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

      10. add-sqr-sqrt -0.0%

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

      11. sqrt-unprod 71.6%

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

      12. sqr-neg 71.6%

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

      13. sqrt-unprod 78.4%

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

      14. add-sqr-sqrt 78.4%

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

      15. frac-2neg 78.4%

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

   8. Applied egg-rr 78.4%

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

      1. unpow2 78.4%

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

      2. associate-/r/ 79.4%

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

      3. unpow2 79.4%

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

   10. Simplified 79.4%

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

4. Final simplification 89.0%

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

Alternative 5: 90.3% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20.0)
   (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0))))
   (/ 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 = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	} 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 = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    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(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s))) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * 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(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	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 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. Taylor expanded in x around 0 94.2%

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

      1. expm1-log1p-u 94.2%

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

      2. expm1-udef 94.2%

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

      3. log1p-udef 94.2%

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

      4. add-exp-log 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. associate--l+ 94.2%

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

      2. sub-neg 94.2%

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

      3. +-commutative 94.2%

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

      4. metadata-eval 94.2%

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

   8. Simplified 94.2%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 71.6%

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

      1. mul-1-neg 71.6%

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

      2. unsub-neg 71.6%

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

      3. unpow2 71.6%

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

      4. unpow2 71.6%

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

      5. times-frac 65.5%

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

   4. Simplified 65.5%

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

      1. clear-num 65.5%

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

      2. frac-times 67.4%

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

      3. *-un-lft-identity 67.4%

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

   6. Applied egg-rr 67.4%

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

      1. associate-*l/ 78.4%

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

   8. Applied egg-rr 78.4%

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

4. Final simplification 88.6%

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

Alternative 6: 88.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1000.0)
   (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0))))
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1000.0f) {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	} 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 / s) <= 1000.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    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(Float32(-x) / s) <= Float32(1000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s))) + Float32(-1.0)))));
	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 / s) <= single(1000.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	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 (/.f32 (neg.f32 x) s) < 1e3

   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 99.6%

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

   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 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-udef 92.1%

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

      3. log1p-udef 92.1%

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

      4. add-exp-log 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. associate--l+ 92.2%

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

      2. sub-neg 92.2%

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

      3. +-commutative 92.2%

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

      4. metadata-eval 92.2%

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

   8. Simplified 92.2%

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

   if 1e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unpow2 74.8%

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

      4. unpow2 74.8%

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

      5. times-frac 68.1%

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

   4. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. frac-times 70.1%

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

      3. *-un-lft-identity 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in x around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. unpow2 74.8%

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

   9. Simplified 74.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 7: 89.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (* 2.0 (/ (* s s) (* x x)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * ((s * s) / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. unsub-neg 70.3%

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

      3. unpow2 70.3%

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

      4. unpow2 70.3%

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

      5. times-frac 64.4%

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

   4. Simplified 64.4%

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

   5. Taylor expanded in x around inf 69.2%

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

      1. unpow2 69.2%

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

      2. unpow2 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 8: 89.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ (* 2.0 (* s s)) (* x x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = (2.0f * (s * s)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = (2.0e0 * (s * s)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(s * s)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = (single(2.0) * (s * s)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. unsub-neg 70.3%

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

      3. unpow2 70.3%

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

      4. unpow2 70.3%

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

      5. times-frac 64.4%

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

   4. Simplified 64.4%

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

      1. clear-num 64.4%

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

      2. frac-times 66.3%

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

      3. *-un-lft-identity 66.3%

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

   6. Applied egg-rr 66.3%

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

   7. Taylor expanded in x around inf 69.2%

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

      1. associate-*r/ 69.2%

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

      2. unpow2 69.2%

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

      3. unpow2 69.2%

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

   9. Simplified 69.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]

Alternative 9: 88.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1000.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1000.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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 / s) <= 1000.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    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(Float32(-x) / s) <= Float32(1000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(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 / s) <= single(1000.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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 (/.f32 (neg.f32 x) s) < 1e3

   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 99.6%

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

   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 92.1%

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

   if 1e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unpow2 74.8%

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

      4. unpow2 74.8%

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

      5. times-frac 68.1%

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

   4. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. frac-times 70.1%

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

      3. *-un-lft-identity 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in x around inf 74.8%

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

      1. *-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. unpow2 74.8%

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

   9. Simplified 74.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 10: 89.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.9999999996399175e-23)
   (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0))))
   (/ 1.0 (+ 2.0 (- (/ (* x x) (* s s)) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.9999999996399175e-23f) {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	} else {
		tmp = 1.0f / (2.0f + (((x * x) / (s * 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 <= 1.9999999996399175e-23) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    else
        tmp = 1.0e0 / (2.0e0 + (((x * x) / (s * s)) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s))) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * x) / Float32(s * s)) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.9999999996399175e-23))
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	else
		tmp = single(1.0) / (single(2.0) + (((x * x) / (s * s)) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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. Taylor expanded in x around 0 90.8%

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

      1. expm1-log1p-u 90.7%

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

      2. expm1-udef 90.7%

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

      3. log1p-udef 90.8%

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

      4. add-exp-log 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. associate--l+ 90.8%

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

      2. sub-neg 90.8%

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

      3. +-commutative 90.8%

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

      4. metadata-eval 90.8%

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

   8. Simplified 90.8%

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

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

   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. Taylor expanded in x around 0 12.5%

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

   5. Taylor expanded in x around 0 78.4%

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

      1. unpow2 78.4%

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

      2. unpow2 78.4%

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

      3. mul-1-neg 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(\frac{x \cdot x}{s \cdot s} - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 11: 75.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0)
     (- 1.0 (/ s x))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. +-commutative 94.9%

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

      2. neg-mul-1 94.9%

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

      3. unsub-neg 94.9%

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

   7. Simplified 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 34.7%

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

      1. mul-1-neg 34.7%

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

      2. unsub-neg 34.7%

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

   4. Simplified 34.7%

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

   5. Taylor expanded in x around inf 34.7%

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

      1. neg-mul-1 34.7%

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

      2. distribute-neg-frac 34.7%

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

   7. Simplified 34.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 12: 75.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 2.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(2.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. *-commutative 97.4%

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

   4. Simplified 97.4%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 34.7%

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

      1. mul-1-neg 34.7%

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

      2. unsub-neg 34.7%

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

   4. Simplified 34.7%

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

   5. Taylor expanded in x around inf 34.7%

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

      1. neg-mul-1 34.7%

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

      2. distribute-neg-frac 34.7%

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

   7. Simplified 34.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 13: 73.5% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0) (- 1.0 (/ s x)) (if (<= t_0 2.0) 0.5 (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. +-commutative 94.9%

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

      2. neg-mul-1 94.9%

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

      3. unsub-neg 94.9%

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

   7. Simplified 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 91.5%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 34.7%

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

      1. mul-1-neg 34.7%

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

      2. unsub-neg 34.7%

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

   4. Simplified 34.7%

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

   5. Taylor expanded in x around inf 34.7%

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

      1. neg-mul-1 34.7%

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

      2. distribute-neg-frac 34.7%

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

   7. Simplified 34.7%

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

4. Final simplification 72.2%

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

Alternative 14: 86.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{\frac{x}{s} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1000.0)
   (/ 1.0 (+ 1.0 (/ -1.0 (+ (/ x s) -1.0))))
   (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1000.0f) {
		tmp = 1.0f / (1.0f + (-1.0f / ((x / s) + -1.0f)));
	} 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) <= 1000.0e0) then
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((x / s) + (-1.0e0))))
    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(1000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(x / s) + Float32(-1.0)))));
	else
		tmp = Float32(Float32(Float32(2.0) * 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(1000.0))
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / ((x / s) + single(-1.0))));
	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) < 1e3

   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 99.6%

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

   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 92.1%

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

      1. frac-2neg 92.1%

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

      2. metadata-eval 92.1%

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

      3. div-inv 92.1%

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

      4. +-commutative 92.1%

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

      5. distribute-neg-in 92.1%

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

      6. distribute-frac-neg 92.1%

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

      7. add-sqr-sqrt 17.3%

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

      8. sqrt-unprod 89.8%

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

      9. sqr-neg 89.8%

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

      10. sqrt-unprod 73.9%

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

      11. add-sqr-sqrt 90.4%

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

      12. metadata-eval 90.4%

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

   6. Applied egg-rr 90.4%

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

      1. metadata-eval 90.4%

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

      2. sub-neg 90.4%

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

      3. associate-*r/ 90.4%

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

      4. metadata-eval 90.4%

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

      5. sub-neg 90.4%

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

      6. metadata-eval 90.4%

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

   8. Simplified 90.4%

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

   if 1e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unpow2 74.8%

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

      4. unpow2 74.8%

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

      5. times-frac 68.1%

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

   4. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. frac-times 70.1%

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

      3. *-un-lft-identity 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in x around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. unpow2 73.7%

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

      3. unpow2 73.7%

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

   9. Simplified 73.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{\frac{x}{s} + -1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]

Alternative 15: 88.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1000.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1000.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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) <= 1000.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    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(1000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(2.0) * 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(1000.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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) < 1e3

   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 99.6%

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

   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 92.1%

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

   if 1e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 74.8%

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

      1. mul-1-neg 74.8%

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

      2. unsub-neg 74.8%

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

      3. unpow2 74.8%

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

      4. unpow2 74.8%

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

      5. times-frac 68.1%

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

   4. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. frac-times 70.1%

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

      3. *-un-lft-identity 70.1%

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

   6. Applied egg-rr 70.1%

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

   7. Taylor expanded in x around inf 73.7%

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

      1. associate-*r/ 73.7%

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

      2. unpow2 73.7%

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

      3. unpow2 73.7%

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

   9. Simplified 73.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1000:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]

Alternative 16: 75.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -4.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-4.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.9%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 59.2%

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

      1. mul-1-neg 59.2%

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

      2. unsub-neg 59.2%

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

   4. Simplified 59.2%

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

4. Final simplification 73.3%

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

Alternative 17: 67.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8)
   (/ s x)
   (if (<= x 3.999999935100636e-17) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		tmp = s / x;
	} else if (x <= 3.999999935100636e-17f) {
		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 <= (-1.999999987845058e-8)) then
        tmp = s / x
    else if (x <= 3.999999935100636e-17) 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(-1.999999987845058e-8))
		tmp = Float32(s / x);
	elseif (x <= Float32(3.999999935100636e-17))
		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(-1.999999987845058e-8))
		tmp = s / x;
	elseif (x <= single(3.999999935100636e-17))
		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 < -1.99999999e-8

   1. Initial program 100.0%

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

   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 42.4%

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

      1. neg-mul-1 42.4%

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

      2. distribute-neg-frac 42.4%

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

   7. Simplified 42.4%

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

      1. associate-/r/ 39.9%

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

      2. add-sqr-sqrt 39.9%

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

      3. sqrt-unprod 53.3%

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

      4. sqr-neg 53.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 39.9%

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

   9. Applied egg-rr 39.9%

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

   10. Taylor expanded in x around 0 39.9%

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

   if -1.99999999e-8 < x < 3.99999994e-17

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 66.8%

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

   if 3.99999994e-17 < 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 95.6%

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

   5. Taylor expanded in x around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. neg-mul-1 93.7%

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

      3. unsub-neg 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{elif}\;x \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 18: 69.2% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8)
   (/ 1.0 (/ x s))
   (if (<= x 3.999999935100636e-17) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 3.999999935100636e-17f) {
		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 <= (-1.999999987845058e-8)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 3.999999935100636e-17) 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(-1.999999987845058e-8))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(3.999999935100636e-17))
		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(-1.999999987845058e-8))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(3.999999935100636e-17))
		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 < -1.99999999e-8

   1. Initial program 100.0%

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

   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 42.4%

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

      1. neg-mul-1 42.4%

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

      2. distribute-neg-frac 42.4%

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

   7. Simplified 42.4%

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

      1. clear-num 39.9%

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

      2. add-sqr-sqrt 39.9%

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

      3. sqrt-unprod 53.3%

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

      4. sqr-neg 53.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 39.9%

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

      7. clear-num 42.4%

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

      8. inv-pow 42.4%

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

   9. Applied egg-rr 42.4%

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

       1. unpow-1 42.4%

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

   11. Simplified 42.4%

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

   if -1.99999999e-8 < x < 3.99999994e-17

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 66.8%

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

   if 3.99999994e-17 < 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 95.6%

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

   5. Taylor expanded in x around inf 93.7%

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

      1. +-commutative 93.7%

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

      2. neg-mul-1 93.7%

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

      3. unsub-neg 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 3.999999935100636 \cdot 10^{-17}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 19: 45.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		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 <= (-1.999999987845058e-8)) 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(-1.999999987845058e-8))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.999999987845058e-8))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-8

   1. Initial program 100.0%

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

   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 42.4%

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

      1. neg-mul-1 42.4%

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

      2. distribute-neg-frac 42.4%

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

   7. Simplified 42.4%

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

      1. associate-/r/ 39.9%

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

      2. add-sqr-sqrt 39.9%

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

      3. sqrt-unprod 53.3%

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

      4. sqr-neg 53.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 39.9%

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

   9. Applied egg-rr 39.9%

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

   10. Taylor expanded in x around 0 39.9%

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

   if -1.99999999e-8 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 47.1%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 20: 34.8% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 35.6%

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

3. Final simplification 35.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023208 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))