Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 22.1s

Alternatives: 19

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf -6631085370769408)
  2. reg1 = (bf #e5.992736498373408092640068303499707800176e-31)

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.4× 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(x / Float32(-s))))))
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
5. 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-expm1-u 99.8%

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

   4. log1p-define 99.9%

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

   5. expm1-log1p-u 99.9%

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

   6. rec-exp 99.9%

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

   7. distribute-neg-frac2 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \sqrt[3]{e^{\frac{x}{s} \cdot -3}}} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

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

   1. add-cbrt-cube 99.8%

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

   2. pow1/3 99.8%

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

   3. pow3 99.8%

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

   4. inv-pow 99.8%

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

   5. pow-pow 99.8%

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

   6. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow1/3 99.8%

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

8. Applied egg-rr 99.8%

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

   1. exp-prod 99.9%

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

10. Simplified 99.9%

    \[\leadsto \frac{1}{1 + \color{blue}{\sqrt[3]{e^{\frac{x}{s} \cdot -3}}}} \]
11. Add Preprocessing

Alternative 3: 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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + e^{\frac{x}{-s}}} \]
4. Add Preprocessing

Alternative 4: 82.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -49999998976.0)
   (/ 1.0 (/ x s))
   (if (<= x -2.9999999105145657e-35)
     (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ (/ x s) 2.0)))
     (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -49999998976.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= -2.9999999105145657e-35f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-49999998976.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= (-2.9999999105145657e-35)) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-49999998976.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(-2.9999999105145657e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-49999998976.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(-2.9999999105145657e-35))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -49999999000

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.4%

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

      1. neg-mul-1 78.4%

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

      2. unsub-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in x around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   8. Simplified 72.5%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 78.4%

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

      5. sqr-neg 78.4%

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

      6. sqrt-unprod 78.4%

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

      7. add-sqr-sqrt 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. unpow-1 78.4%

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

   12. Simplified 78.4%

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

   if -49999999000 < x < -2.99999991e-35

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 38.6%

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

      1. neg-mul-1 38.6%

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

      2. unsub-neg 38.6%

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

   5. Simplified 38.6%

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

      1. sub-neg 38.6%

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

      2. flip-+ 62.6%

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

      3. metadata-eval 62.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      4. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      5. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \left(-\frac{x}{s}\right)}} \]

      6. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \color{blue}{\frac{x}{-s}}}} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}} \]

      2. distribute-frac-neg2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \frac{x}{-s}}} \]

      3. sqr-neg 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}{2 - \frac{x}{-s}}} \]

      4. frac-2neg 62.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \color{blue}{\frac{-x}{-s}}}{2 - \frac{x}{-s}}} \]

      5. associate-*r/ 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{s} \cdot \left(-x\right)}{-s}}}{2 - \frac{x}{-s}}} \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{2 - \frac{x}{-s}}} \]

      7. sqrt-unprod 74.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{2 - \frac{x}{-s}}} \]

      8. sqr-neg 74.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\sqrt{\color{blue}{s \cdot s}}}}{2 - \frac{x}{-s}}} \]

      9. sqrt-unprod 61.9%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{2 - \frac{x}{-s}}} \]

      10. add-sqr-sqrt 61.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{s}}}{2 - \frac{x}{-s}}} \]

   9. Applied egg-rr 61.9%

      \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{s} \cdot \left(-x\right)}{s}}}{2 - \frac{x}{-s}}} \]
   10. Step-by-step derivation

       1. *-commutative 61.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\left(-x\right) \cdot \frac{x}{s}}}{s}}{2 - \frac{x}{-s}}} \]

       2. *-un-lft-identity 61.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{\left(-x\right) \cdot \frac{x}{s}}{\color{blue}{1 \cdot s}}}{2 - \frac{x}{-s}}} \]

       3. times-frac 70.3%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{-x}{1} \cdot \frac{\frac{x}{s}}{s}}}{2 - \frac{x}{-s}}} \]

       4. add-sqr-sqrt 70.3%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       5. sqrt-unprod 67.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       6. sqr-neg 67.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{\sqrt{\color{blue}{x \cdot x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       7. sqrt-unprod -0.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       8. add-sqr-sqrt 71.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{x}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

   11. Applied egg-rr 71.0%

       \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{1} \cdot \frac{\frac{x}{s}}{s}}}{2 - \frac{x}{-s}}} \]

   if -2.99999991e-35 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. expm1-log1p-u 93.6%

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

      2. log1p-define 93.6%

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

      3. expm1-undefine 93.6%

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

      4. add-exp-log 93.6%

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

      5. +-commutative 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate--l+ 93.7%

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

       2. sub-neg 93.7%

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

       3. +-commutative 93.7%

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

       4. metadata-eval 93.7%

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

   11. Simplified 93.7%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -49999998976.0)
   (/ 1.0 (/ x s))
   (if (<= x -2.9999999105145657e-35)
     (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ (/ x s) 2.0)))
     (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -49999998976.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= -2.9999999105145657e-35f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-49999998976.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= (-2.9999999105145657e-35)) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-49999998976.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(-2.9999999105145657e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-49999998976.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(-2.9999999105145657e-35))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -49999999000

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.4%

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

      1. neg-mul-1 78.4%

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

      2. unsub-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in x around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   8. Simplified 72.5%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 78.4%

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

      5. sqr-neg 78.4%

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

      6. sqrt-unprod 78.4%

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

      7. add-sqr-sqrt 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. unpow-1 78.4%

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

   12. Simplified 78.4%

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

   if -49999999000 < x < -2.99999991e-35

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 38.6%

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

      1. neg-mul-1 38.6%

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

      2. unsub-neg 38.6%

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

   5. Simplified 38.6%

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

      1. sub-neg 38.6%

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

      2. flip-+ 62.6%

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

      3. metadata-eval 62.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      4. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      5. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \left(-\frac{x}{s}\right)}} \]

      6. distribute-neg-frac2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \color{blue}{\frac{x}{-s}}}} \]

   7. Applied egg-rr 62.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}} \]

      2. distribute-frac-neg2 62.6%

         \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \frac{x}{-s}}} \]

      3. sqr-neg 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}{2 - \frac{x}{-s}}} \]

      4. clear-num 62.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1}{\frac{s}{x}}} \cdot \frac{x}{s}}{2 - \frac{x}{-s}}} \]

      5. frac-times 66.1%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1 \cdot x}{\frac{s}{x} \cdot s}}}{2 - \frac{x}{-s}}} \]

      6. *-un-lft-identity 66.1%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{x}}{\frac{s}{x} \cdot s}}{2 - \frac{x}{-s}}} \]

   9. Applied egg-rr 66.1%

      \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{\frac{s}{x} \cdot s}}}{2 - \frac{x}{-s}}} \]

   if -2.99999991e-35 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. expm1-log1p-u 93.6%

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

      2. log1p-define 93.6%

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

      3. expm1-undefine 93.6%

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

      4. add-exp-log 93.6%

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

      5. +-commutative 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate--l+ 93.7%

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

       2. sub-neg 93.7%

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

       3. +-commutative 93.7%

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

       4. metadata-eval 93.7%

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

   11. Simplified 93.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -49999998976.0)
   (/ 1.0 (/ x s))
   (if (<= x -1.999999943436137e-9)
     (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (/ x s)))
     (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -49999998976.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= -1.999999943436137e-9f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (x / s));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-49999998976.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= (-1.999999943436137e-9)) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-49999998976.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(-1.999999943436137e-9))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(x / s)));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-49999998976.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(-1.999999943436137e-9))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -49999999000

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.4%

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

      1. neg-mul-1 78.4%

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

      2. unsub-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in x around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   8. Simplified 72.5%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 78.4%

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

      5. sqr-neg 78.4%

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

      6. sqrt-unprod 78.4%

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

      7. add-sqr-sqrt 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. unpow-1 78.4%

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

   12. Simplified 78.4%

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

   if -49999999000 < x < -1.99999994e-9

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 17.2%

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

      1. neg-mul-1 17.2%

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

      2. unsub-neg 17.2%

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

   5. Simplified 17.2%

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

      1. sub-neg 17.2%

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

      2. flip-+ 61.6%

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

      3. metadata-eval 61.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      4. distribute-neg-frac2 61.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      5. distribute-neg-frac2 61.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \left(-\frac{x}{s}\right)}} \]

      6. distribute-neg-frac2 61.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \color{blue}{\frac{x}{-s}}}} \]

   7. Applied egg-rr 61.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]

   8. Taylor expanded in x around inf 57.6%

      \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\color{blue}{\frac{x}{s}}}} \]

   if -1.99999994e-9 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 86.3%

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

      1. +-commutative 86.3%

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

   7. Simplified 86.3%

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

      1. expm1-log1p-u 86.3%

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

      2. log1p-define 86.3%

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

      3. expm1-undefine 86.3%

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

      4. add-exp-log 86.3%

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

      5. +-commutative 86.3%

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

   9. Applied egg-rr 86.3%

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

       1. associate--l+ 86.4%

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

       2. sub-neg 86.4%

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

       3. +-commutative 86.4%

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

       4. metadata-eval 86.4%

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

   11. Simplified 86.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{4 + \frac{x \cdot \frac{x}{s}}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -49999998976.0)
   (/ 1.0 (/ x s))
   (if (<= x -1.0000000116860974e-7)
     (/ 1.0 (/ (+ 4.0 (/ (* x (/ x s)) s)) (/ x s)))
     (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -49999998976.0f) {
		tmp = 1.0f / (x / s);
	} else if (x <= -1.0000000116860974e-7f) {
		tmp = 1.0f / ((4.0f + ((x * (x / s)) / s)) / (x / s));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-49999998976.0e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= (-1.0000000116860974e-7)) then
        tmp = 1.0e0 / ((4.0e0 + ((x * (x / s)) / s)) / (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-49999998976.0))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(-1.0000000116860974e-7))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(x * Float32(x / s)) / s)) / Float32(x / s)));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-49999998976.0))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(-1.0000000116860974e-7))
		tmp = single(1.0) / ((single(4.0) + ((x * (x / s)) / s)) / (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -49999999000

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 78.4%

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

      1. neg-mul-1 78.4%

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

      2. unsub-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in x around inf 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   8. Simplified 72.5%

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

      1. clear-num 78.4%

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

      2. inv-pow 78.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 78.4%

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

      5. sqr-neg 78.4%

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

      6. sqrt-unprod 78.4%

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

      7. add-sqr-sqrt 78.4%

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

   10. Applied egg-rr 78.4%

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

       1. unpow-1 78.4%

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

   12. Simplified 78.4%

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

   if -49999999000 < x < -1.00000001e-7

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 15.3%

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

      1. neg-mul-1 15.3%

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

      2. unsub-neg 15.3%

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

   5. Simplified 15.3%

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

      1. sub-neg 15.3%

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

      2. flip-+ 62.2%

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

      3. metadata-eval 62.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      4. distribute-neg-frac2 62.2%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      5. distribute-neg-frac2 62.2%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \left(-\frac{x}{s}\right)}} \]

      6. distribute-neg-frac2 62.2%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \color{blue}{\frac{x}{-s}}}} \]

   7. Applied egg-rr 62.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]
   8. Step-by-step derivation

      1. distribute-frac-neg2 62.2%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}} \]

      2. distribute-frac-neg2 62.2%

         \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \frac{x}{-s}}} \]

      3. sqr-neg 62.2%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}{2 - \frac{x}{-s}}} \]

      4. frac-2neg 62.2%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \color{blue}{\frac{-x}{-s}}}{2 - \frac{x}{-s}}} \]

      5. associate-*r/ 62.2%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{s} \cdot \left(-x\right)}{-s}}}{2 - \frac{x}{-s}}} \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{2 - \frac{x}{-s}}} \]

      7. sqrt-unprod 61.8%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{2 - \frac{x}{-s}}} \]

      8. sqr-neg 61.8%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\sqrt{\color{blue}{s \cdot s}}}}{2 - \frac{x}{-s}}} \]

      9. sqrt-unprod 61.8%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{2 - \frac{x}{-s}}} \]

      10. add-sqr-sqrt 61.8%

          \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{\color{blue}{s}}}{2 - \frac{x}{-s}}} \]

   9. Applied egg-rr 61.8%

      \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{s} \cdot \left(-x\right)}{s}}}{2 - \frac{x}{-s}}} \]

   10. Taylor expanded in x around inf 60.1%

       \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{s} \cdot \left(-x\right)}{s}}{\color{blue}{\frac{x}{s}}}} \]

   if -1.00000001e-7 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 85.1%

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

      1. +-commutative 85.1%

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

   7. Simplified 85.1%

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

      1. expm1-log1p-u 85.1%

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

      2. log1p-define 85.1%

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

      3. expm1-undefine 85.1%

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

      4. add-exp-log 85.1%

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

      5. +-commutative 85.1%

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

   9. Applied egg-rr 85.1%

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

       1. associate--l+ 85.1%

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

       2. sub-neg 85.1%

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

       3. +-commutative 85.1%

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

       4. metadata-eval 85.1%

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

   11. Simplified 85.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -49999998976:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{4 + \frac{x \cdot \frac{x}{s}}{s}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.9999999105145657e-35)
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))
   (/ 1.0 (+ 1.0 (+ 1.0 (+ (/ 1.0 (+ 1.0 (/ x s))) -1.0))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.9999999105145657e-35f) {
		tmp = 1.0f / (2.0f + (x * (-1.0f / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((1.0f / (1.0f + (x / s))) + -1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.9999999105145657e-35)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((-1.0e0) / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((1.0e0 / (1.0e0 + (x / s))) + (-1.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.9999999105145657e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(-1.0) / s))));
	else
		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)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.9999999105145657e-35))
		tmp = single(1.0) / (single(2.0) + (x * (single(-1.0) / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((single(1.0) / (single(1.0) + (x / s))) + single(-1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.99999991e-35

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

      2. unsub-neg 54.0%

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

   5. Simplified 54.0%

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

      1. clear-num 54.0%

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

      2. associate-/r/ 54.0%

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

   7. Applied egg-rr 54.0%

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

   if -2.99999991e-35 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

      1. expm1-log1p-u 93.6%

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

      2. log1p-define 93.6%

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

      3. expm1-undefine 93.6%

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

      4. add-exp-log 93.6%

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

      5. +-commutative 93.6%

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

   9. Applied egg-rr 93.6%

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

       1. associate--l+ 93.7%

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

       2. sub-neg 93.7%

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

       3. +-commutative 93.7%

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

       4. metadata-eval 93.7%

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

   11. Simplified 93.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \left(\frac{1}{1 + \frac{x}{s}} + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291)
   (/ -1.0 (/ x s))
   (if (<= x 4.00000018325482e-18)
     (+ 0.5 (/ (* x 0.25) s))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		tmp = -1.0f / (x / s);
	} else if (x <= 4.00000018325482e-18f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.004999999888241291e0)) then
        tmp = (-1.0e0) / (x / s)
    else if (x <= 4.00000018325482e-18) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.004999999888241291))
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	elseif (x <= Float32(4.00000018325482e-18))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = single(-1.0) / (x / s);
	elseif (x <= single(4.00000018325482e-18))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00499999989

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

      2. unsub-neg 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around inf 54.4%

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

      1. mul-1-neg 54.4%

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

      2. distribute-frac-neg2 54.4%

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

   8. Simplified 54.4%

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

   if -0.00499999989 < x < 4.00000018e-18

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.1%

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

      1. associate-*r/ 62.1%

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

   5. Simplified 62.1%

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

   if 4.00000018e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.6%

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

      1. +-commutative 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in x around inf 90.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291)
   (/ -1.0 (/ x s))
   (if (<= x 4.000000014509975e-15) (+ 0.5 (/ (* x 0.25) s)) (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		tmp = -1.0f / (x / s);
	} else if (x <= 4.000000014509975e-15f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.004999999888241291e0)) then
        tmp = (-1.0e0) / (x / s)
    else if (x <= 4.000000014509975e-15) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.004999999888241291))
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	elseif (x <= Float32(4.000000014509975e-15))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = single(-1.0) / (x / s);
	elseif (x <= single(4.000000014509975e-15))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00499999989

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

      2. unsub-neg 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around inf 54.4%

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

      1. mul-1-neg 54.4%

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

      2. distribute-frac-neg2 54.4%

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

   8. Simplified 54.4%

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

   if -0.00499999989 < x < 4.00000001e-15

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.2%

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

      1. associate-*r/ 61.2%

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

   5. Simplified 61.2%

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

   if 4.00000001e-15 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in x around inf 91.9%

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

      1. neg-mul-1 91.9%

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

      2. unsub-neg 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.9999999105145657e-35)
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.9999999105145657e-35f) {
		tmp = 1.0f / (2.0f + (x * (-1.0f / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.9999999105145657e-35)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((-1.0e0) / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.9999999105145657e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(-1.0) / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.9999999105145657e-35))
		tmp = single(1.0) / (single(2.0) + (x * (single(-1.0) / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.99999991e-35

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.0%

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

      1. neg-mul-1 54.0%

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

      2. unsub-neg 54.0%

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

   5. Simplified 54.0%

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

      1. clear-num 54.0%

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

      2. associate-/r/ 54.0%

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

   7. Applied egg-rr 54.0%

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

   if -2.99999991e-35 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

      1. +-commutative 93.6%

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

   7. Simplified 93.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9999999105145657 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291)
   (/ -1.0 (/ x s))
   (if (<= x 4.000000014509975e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		tmp = -1.0f / (x / s);
	} else if (x <= 4.000000014509975e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.004999999888241291e0)) then
        tmp = (-1.0e0) / (x / s)
    else if (x <= 4.000000014509975e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.004999999888241291))
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	elseif (x <= Float32(4.000000014509975e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = single(-1.0) / (x / s);
	elseif (x <= single(4.000000014509975e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00499999989

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

      2. unsub-neg 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around inf 54.4%

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

      1. mul-1-neg 54.4%

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

      2. distribute-frac-neg2 54.4%

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

   8. Simplified 54.4%

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

   if -0.00499999989 < x < 4.00000001e-15

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 59.8%

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

   if 4.00000001e-15 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in x around inf 91.9%

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

      1. neg-mul-1 91.9%

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

      2. unsub-neg 91.9%

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

   10. Simplified 91.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.009999999776482582)
   (/ 1.0 (/ x s))
   (if (<= x 4.000000014509975e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.009999999776482582f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 4.000000014509975e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.009999999776482582e0)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 4.000000014509975e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.009999999776482582))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(4.000000014509975e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.009999999776482582))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(4.000000014509975e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00999999978

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 56.1%

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

      1. neg-mul-1 56.1%

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

      2. unsub-neg 56.1%

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

   5. Simplified 56.1%

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

   6. Taylor expanded in x around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. neg-mul-1 51.6%

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

   8. Simplified 51.6%

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

      1. clear-num 56.1%

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

      2. inv-pow 56.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 65.2%

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

      5. sqr-neg 65.2%

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

      6. sqrt-unprod 56.1%

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

      7. add-sqr-sqrt 56.1%

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

   10. Applied egg-rr 56.1%

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

       1. unpow-1 56.1%

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

   12. Simplified 56.1%

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

   if -0.00999999978 < x < 4.00000001e-15

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 58.2%

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

   if 4.00000001e-15 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in x around inf 91.9%

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

      1. neg-mul-1 91.9%

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

      2. unsub-neg 91.9%

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

   10. Simplified 91.9%

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

Alternative 14: 67.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 4.000000014509975 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291)
   (/ (- s) x)
   (if (<= x 4.000000014509975e-15) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		tmp = -s / x;
	} else if (x <= 4.000000014509975e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.004999999888241291e0)) then
        tmp = -s / x
    else if (x <= 4.000000014509975e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.004999999888241291))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(4.000000014509975e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = -s / x;
	elseif (x <= single(4.000000014509975e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00499999989

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

      2. unsub-neg 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. neg-mul-1 50.1%

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

   8. Simplified 50.1%

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

   if -0.00499999989 < x < 4.00000001e-15

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 59.8%

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

   if 4.00000001e-15 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.8%

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

      1. +-commutative 95.8%

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

   7. Simplified 95.8%

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

   8. Taylor expanded in x around inf 91.9%

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

      1. neg-mul-1 91.9%

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

      2. unsub-neg 91.9%

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

   10. Simplified 91.9%

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

Alternative 15: 69.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.00000018325482e-18)
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))
   (/ 1.0 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.00000018325482e-18f) {
		tmp = 1.0f / (-1.0f + (3.0f - (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.00000018325482e-18) then
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.00000018325482e-18))
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.00000018325482e-18))
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000018e-18

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 57.6%

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

      1. neg-mul-1 57.6%

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

      2. unsub-neg 57.6%

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

   5. Simplified 57.6%

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

      1. expm1-log1p-u 57.4%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]

   7. Applied egg-rr 57.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 57.4%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} - 1}} \]

      2. sub-neg 57.4%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} + \left(-1\right)}} \]

      3. log1p-undefine 57.4%

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

      4. rem-exp-log 57.7%

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

      5. associate-+r- 57.7%

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

      6. metadata-eval 57.7%

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

      7. metadata-eval 57.7%

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

   9. Simplified 57.7%

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

   if 4.00000018e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.6%

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

      1. +-commutative 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in x around inf 90.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.00000018325482 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.00000018325482e-18)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.00000018325482e-18f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + (s / x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.00000018325482e-18) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.00000018325482e-18))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.00000018325482e-18))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.00000018e-18

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 57.6%

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

      1. neg-mul-1 57.6%

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

      2. unsub-neg 57.6%

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

   5. Simplified 57.6%

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

   if 4.00000018e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 94.6%

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

      1. +-commutative 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in x around inf 90.3%

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

Alternative 17: 46.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.004999999888241291e0)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.004999999888241291))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.004999999888241291))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00499999989

   1. Initial program 99.9%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.4%

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

      1. neg-mul-1 54.4%

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

      2. unsub-neg 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. neg-mul-1 50.1%

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

   8. Simplified 50.1%

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

   if -0.00499999989 < x

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 45.3%

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

Alternative 18: 46.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.009999999776482582:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.009999999776482582) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.009999999776482582f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.009999999776482582e0)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.009999999776482582))
		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(-0.009999999776482582))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00999999978

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 56.1%

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

      1. neg-mul-1 56.1%

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

      2. unsub-neg 56.1%

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

   5. Simplified 56.1%

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

   6. Taylor expanded in x around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. neg-mul-1 51.6%

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

   8. Simplified 51.6%

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

      1. div-inv 51.6%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 62.0%

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

      4. sqr-neg 62.0%

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

      5. sqrt-unprod 51.6%

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

      6. add-sqr-sqrt 51.6%

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

   10. Applied egg-rr 51.6%

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

       1. associate-*r/ 51.6%

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

       2. *-rgt-identity 51.6%

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

   12. Simplified 51.6%

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

   if -0.00999999978 < x

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 44.7%

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

Alternative 19: 35.0% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 33.6%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))