Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 12.4s

Alternatives: 23

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{0 - \frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- 0.0 (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((0.0f - (x / s)))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(0.0) - Float32(x / s))))))
end

Derivation

1. Initial program 99.9%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. log-pow N/A

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

   5. inv-pow N/A

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

   6. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\log \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

   7. log-rec N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\log \left(1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

   9. log1p-define N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   10. log1p-lowering-log1p.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   11. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right)\right) \]

   12. distribute-frac-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right)\right) \]

   13. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right)\right)\right) \]

   15. neg-lowering-neg.f32 99.9%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{0 - \frac{x}{s}}\right)} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

   2. associate--r- N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

   4. associate--r+ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

   8. exp-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

   12. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

   13. /-lowering-/.f32 99.9%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{\color{blue}{1 - \frac{-1}{e^{\frac{x}{s}}}}} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{0 - \frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (- 0.0 (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((0.0f - (x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((0.0e0 - (x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((single(0.0) - (x / s))));
end

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + e^{0 - \frac{x}{s}}} \]
4. Add Preprocessing

Alternative 4: 93.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{\frac{x}{s}}{s \cdot s} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x + \frac{0.5 \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{0.16666666666666666}{s}}{s}}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (+ (* -0.16666666666666666 (/ (/ x s) (* s s))) (/ 0.5 (* s s))))
       (/ -1.0 s)))))
   (/
    1.0
    (+
     1.0
     (/
      -1.0
      (-
       -1.0
       (/
        (+
         x
         (/ (+ (* 0.5 (* x x)) (* (* x (* x x)) (/ 0.16666666666666666 s))) s))
        s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((x * ((-0.16666666666666666f * ((x / s) / (s * s))) + (0.5f / (s * s)))) + (-1.0f / s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - ((x + (((0.5f * (x * x)) + ((x * (x * x)) * (0.16666666666666666f / s))) / s)) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * (((-0.16666666666666666e0) * ((x / s) / (s * s))) + (0.5e0 / (s * s)))) + ((-1.0e0) / s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - ((x + (((0.5e0 * (x * x)) + ((x * (x * x)) * (0.16666666666666666e0 / s))) / s)) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(x / s) / Float32(s * s))) + Float32(Float32(0.5) / Float32(s * s)))) + Float32(Float32(-1.0) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(Float32(x + Float32(Float32(Float32(Float32(0.5) * Float32(x * x)) + Float32(Float32(x * Float32(x * x)) * Float32(Float32(0.16666666666666666) / s))) / s)) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((x * ((single(-0.16666666666666666) * ((x / s) / (s * s))) + (single(0.5) / (s * s)))) + (single(-1.0) / s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - ((x + (((single(0.5) * (x * x)) + ((x * (x * x)) * (single(0.16666666666666666) / s))) / s)) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \left(\frac{\frac{x}{s}}{s \cdot s}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\left(\frac{x}{s}\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 90.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 90.8%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\frac{\frac{x}{s}}{s \cdot s}} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)} \]

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + -1 \cdot \frac{\frac{1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

   7. Simplified 96.2%

      \[\leadsto \frac{1}{1 - \frac{-1}{\color{blue}{1 - \frac{\frac{0.5 \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{0.16666666666666666}{s}}{-s} - x}{s}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{\frac{x}{s}}{s \cdot s} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x + \frac{0.5 \cdot \left(x \cdot x\right) + \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{0.16666666666666666}{s}}{s}}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{\frac{x}{s}}{s \cdot s} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (+ (* -0.16666666666666666 (/ (/ x s) (* s s))) (/ 0.5 (* s s))))
       (/ -1.0 s)))))
   (/ 1.0 (+ 1.0 (/ -1.0 (+ -1.0 (/ (- (/ (* (* x x) -0.5) s) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((x * ((-0.16666666666666666f * ((x / s) / (s * s))) + (0.5f / (s * s)))) + (-1.0f / s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (((((x * x) * -0.5f) / s) - x) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * (((-0.16666666666666666e0) * ((x / s) / (s * s))) + (0.5e0 / (s * s)))) + ((-1.0e0) / s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (((((x * x) * (-0.5e0)) / s) - x) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(-0.16666666666666666) * Float32(Float32(x / s) / Float32(s * s))) + Float32(Float32(0.5) / Float32(s * s)))) + Float32(Float32(-1.0) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(x * x) * Float32(-0.5)) / s) - x) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((x * ((single(-0.16666666666666666) * ((x / s) / (s * s))) + (single(0.5) / (s * s)))) + (single(-1.0) / s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) + (((((x * x) * single(-0.5)) / s) - x) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \left(\frac{\frac{x}{s}}{s \cdot s}\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\left(\frac{x}{s}\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 90.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(\mathsf{/.f32}\left(x, s\right), \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 90.8%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \color{blue}{\frac{\frac{x}{s}}{s \cdot s}} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)} \]

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + -1 \cdot x\right), s\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + \left(\mathsf{neg}\left(x\right)\right)\right), s\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} - x\right), s\right)\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \frac{-1}{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \left(-1 + \frac{1}{2}\right)}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \left(-1 + \frac{1}{2}\right)\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 95.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

   7. Simplified 95.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{\frac{x}{s}}{s \cdot s} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{\frac{0.5}{s} - \frac{\frac{x \cdot 0.16666666666666666}{s}}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 s)
       (* x (/ (- (/ 0.5 s) (/ (/ (* x 0.16666666666666666) s) s)) s))))))
   (/ 1.0 (+ 1.0 (/ -1.0 (+ -1.0 (/ (- (/ (* (* x x) -0.5) s) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (((0.5f / s) - (((x * 0.16666666666666666f) / s) / s)) / s)))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (((((x * x) * -0.5f) / s) - x) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (((0.5e0 / s) - (((x * 0.16666666666666666e0) / s) / s)) / s)))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (((((x * x) * (-0.5e0)) / s) - x) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(Float32(Float32(0.5) / s) - Float32(Float32(Float32(x * Float32(0.16666666666666666)) / s) / s)) / s))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(x * x) * Float32(-0.5)) / s) - x) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (((single(0.5) / s) - (((x * single(0.16666666666666666)) / s) / s)) / s)))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) + (((((x * x) * single(-0.5)) / s) - x) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{-1}{6} \cdot x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{s}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \mathsf{neg}\left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. fmm-undef N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\left(\frac{\frac{1}{2}}{s \cdot s}\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\left(\frac{\frac{-1}{6} \cdot x}{s \cdot s}\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f32 90.0%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 90.0%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{x \cdot -0.16666666666666666}{s \cdot s}}{-s}\right)} + \frac{-1}{s}\right)} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{\frac{1}{2}}{s}}{s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{\frac{1}{2}}{s}}{s} - \left(\mathsf{neg}\left(\frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{s}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{\frac{1}{2}}{s}}{s} - \frac{\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s \cdot s}\right)}{s}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. sub-div N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{\frac{1}{2}}{s} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s \cdot s}\right)\right)}{s}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{\frac{1}{2}}{s} - \left(\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s \cdot s}\right)\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{1}{2}}{s}\right), \left(\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s \cdot s}\right)\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s \cdot s}\right)\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(\mathsf{neg}\left(\frac{\frac{x \cdot \frac{-1}{6}}{s}}{s}\right)\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(\frac{\mathsf{neg}\left(\frac{x \cdot \frac{-1}{6}}{s}\right)}{s}\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      10. distribute-frac-neg2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \left(\frac{\frac{x \cdot \frac{-1}{6}}{\mathsf{neg}\left(s\right)}}{s}\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\left(\frac{x \cdot \frac{-1}{6}}{\mathsf{neg}\left(s\right)}\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      12. frac-2neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\left(\frac{\mathsf{neg}\left(x \cdot \frac{-1}{6}\right)}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\left(\frac{\mathsf{neg}\left(x \cdot \frac{-1}{6}\right)}{s}\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(x \cdot \frac{-1}{6}\right)\right), s\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)\right), s\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\frac{-1}{6}\right)\right)\right), s\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      17. metadata-eval 90.0%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, s\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{1}{6}\right), s\right), s\right)\right), s\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 90.0%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\frac{\frac{0.5}{s} - \frac{\frac{x \cdot 0.16666666666666666}{s}}{s}}{s}} + \frac{-1}{s}\right)} \]

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + -1 \cdot x\right), s\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + \left(\mathsf{neg}\left(x\right)\right)\right), s\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} - x\right), s\right)\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \frac{-1}{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \left(-1 + \frac{1}{2}\right)}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \left(-1 + \frac{1}{2}\right)\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 95.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

   7. Simplified 95.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{\frac{0.5}{s} - \frac{\frac{x \cdot 0.16666666666666666}{s}}{s}}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 s)
       (* x (/ (+ 0.5 (/ (* x -0.16666666666666666) s)) (* s s)))))))
   (/ 1.0 (+ 1.0 (/ -1.0 (+ -1.0 (/ (- (/ (* (* x x) -0.5) s) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((0.5f + ((x * -0.16666666666666666f) / s)) / (s * s))))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (((((x * x) * -0.5f) / s) - x) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((0.5e0 + ((x * (-0.16666666666666666e0)) / s)) / (s * s))))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (((((x * x) * (-0.5e0)) / s) - x) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.16666666666666666)) / s)) / Float32(s * s)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(x * x) * Float32(-0.5)) / s) - x) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((single(0.5) + ((x * single(-0.16666666666666666)) / s)) / (s * s))))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) + (((((x * x) * single(-0.5)) / s) - x) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot \frac{x}{s}\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{\frac{-1}{6} \cdot x}{s}\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 90.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 90.0%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}} + \frac{-1}{s}\right)} \]

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + -1 \cdot x\right), s\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + \left(\mathsf{neg}\left(x\right)\right)\right), s\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} - x\right), s\right)\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \frac{-1}{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \left(-1 + \frac{1}{2}\right)}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \left(-1 + \frac{1}{2}\right)\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 95.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

   7. Simplified 95.5%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \left(x \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/
    1.0
    (+
     2.0
     (* x (+ (/ -1.0 s) (* x (* x (/ -0.16666666666666666 (* s (* s s)))))))))
   (/ 1.0 (+ 1.0 (/ -1.0 (+ -1.0 (/ (- (/ (* (* x x) -0.5) s) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (x * (-0.16666666666666666f / (s * (s * s))))))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (((((x * x) * -0.5f) / s) - x) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (x * ((-0.16666666666666666e0) / (s * (s * s))))))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (((((x * x) * (-0.5e0)) / s) - x) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(x * Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s)))))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(x * x) * Float32(-0.5)) / s) - x) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (x * (single(-0.16666666666666666) / (s * (s * s))))))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) + (((((x * x) * single(-0.5)) / s) - x) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6} \cdot x}{{s}^{3}}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{x \cdot \frac{-1}{6}}{{s}^{3}}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \frac{\frac{-1}{6}}{{s}^{3}}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \frac{\mathsf{neg}\left(\frac{1}{6}\right)}{{s}^{3}}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{6}}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot 1}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\frac{\frac{1}{6} \cdot 1}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\mathsf{neg}\left(\frac{\frac{1}{6}}{{s}^{3}}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{\mathsf{neg}\left(\frac{1}{6}\right)}{{s}^{3}}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6}}{{s}^{3}}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \left({s}^{3}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \left(s \cdot \left(s \cdot s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \left(s \cdot {s}^{2}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(s, \left({s}^{2}\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(s, \left(s \cdot s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 86.6%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 86.6%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)} + \frac{-1}{s}\right)} \]

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + -1 \cdot x\right), s\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + \left(\mathsf{neg}\left(x\right)\right)\right), s\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} - x\right), s\right)\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \frac{-1}{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \left(-1 + \frac{1}{2}\right)}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \left(-1 + \frac{1}{2}\right)\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 95.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

   7. Simplified 95.5%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \left(x \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5 - s}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14)
   (/ 1.0 (/ (* x (* -0.16666666666666666 (* x x))) (* s (* s s))))
   (if (<= x -1.0000000195414814e-25)
     (/ 1.0 (+ 2.0 (* x (/ (- (* x 0.5) s) (* s s)))))
     (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = 1.0f / ((x * (-0.16666666666666666f * (x * x))) / (s * (s * s)));
	} else if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * (((x * 0.5f) - s) / (s * s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = 1.0e0 / ((x * ((-0.16666666666666666e0) * (x * x))) / (s * (s * s)))
    else if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((x * 0.5e0) - s) / (s * s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(-0.16666666666666666) * Float32(x * x))) / Float32(s * Float32(s * s))));
	elseif (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(x * Float32(0.5)) - s) / Float32(s * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = single(1.0) / ((x * (single(-0.16666666666666666) * (x * x))) / (s * (s * s)));
	elseif (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * (((x * single(0.5)) - s) / (s * s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.2%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{-1}{6} \cdot x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{s}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \mathsf{neg}\left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. fmm-undef N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\left(\frac{\frac{1}{2}}{s \cdot s}\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\left(\frac{\frac{-1}{6} \cdot x}{s \cdot s}\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f32 87.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 87.2%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{x \cdot -0.16666666666666666}{s \cdot s}}{-s}\right)} + \frac{-1}{s}\right)} \]
   8. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right) + \color{blue}{x \cdot \frac{-1}{s}}\right)\right)\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right) + x \cdot \frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right) + x \cdot \frac{1}{\mathsf{neg}\left(\color{blue}{s}\right)}\right)\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right) + \frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right) \]

      5. associate-+r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(2 + x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right) + \color{blue}{\frac{x}{\mathsf{neg}\left(s\right)}}\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(2 + x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(2 + x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right) - \color{blue}{\frac{x}{s}}\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(\left(2 + x \cdot \left(x \cdot \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{x \cdot \frac{-1}{6}}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

   9. Applied egg-rr 87.2%

      \[\leadsto \frac{1}{\color{blue}{\left(2 + \frac{0.5 - \frac{x \cdot 0.16666666666666666}{s}}{s \cdot s} \cdot \left(x \cdot x\right)\right) - \frac{x}{s}}} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot {x}^{3}\right), \color{blue}{\left({s}^{3}\right)}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left({x}^{3} \cdot \frac{-1}{6}\right), \left({\color{blue}{s}}^{3}\right)\right)\right) \]

       4. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left({\color{blue}{s}}^{3}\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left({x}^{2} \cdot \frac{-1}{6}\right)\right), \left({\color{blue}{s}}^{3}\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{6}\right)\right), \left({s}^{3}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right), \left({s}^{3}\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \left({s}^{3}\right)\right)\right) \]

       11. cube-mult N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \left(s \cdot {s}^{\color{blue}{2}}\right)\right)\right) \]

       13. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f32}\left(s, \color{blue}{\left({s}^{2}\right)}\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right) \]

       15. *-lowering-*.f32 88.2%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right)\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right) \]

   12. Simplified 88.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(\left(x \cdot x\right) \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}}} \]

   if -1.99999996e-14 < x < -1.00000002e-25

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 81.1%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 99.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 99.1%

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

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1 \cdot s + \frac{1}{2} \cdot x}{{s}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(-1 \cdot s + \frac{1}{2} \cdot x\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x + -1 \cdot s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(s\right)\right)\right), \left({s}^{2}\right)\right)\right)\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x - s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot x\right), s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \left({s}^{2}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 95.4%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Simplified 95.4%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(-0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5 - s}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5 - s}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14)
   (/ (* (* s (* s s)) -6.0) (* x (* x x)))
   (if (<= x -1.0000000195414814e-25)
     (/ 1.0 (+ 2.0 (* x (/ (- (* x 0.5) s) (* s s)))))
     (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = ((s * (s * s)) * -6.0f) / (x * (x * x));
	} else if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * (((x * 0.5f) - s) / (s * s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = ((s * (s * s)) * (-6.0e0)) / (x * (x * x))
    else if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((x * 0.5e0) - s) / (s * s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	elseif (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(x * Float32(0.5)) - s) / Float32(s * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = ((s * (s * s)) * single(-6.0)) / (x * (x * x));
	elseif (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * (((x * single(0.5)) - s) / (s * s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.2%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-6 \cdot {s}^{3}}{\color{blue}{{x}^{3}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-6 \cdot {s}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left({s}^{3} \cdot -6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{3}\right), -6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \left(s \cdot s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot {s}^{2}\right), -6\right), \left({x}^{3}\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2}\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f32 88.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]

   if -1.99999996e-14 < x < -1.00000002e-25

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 81.1%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 99.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 99.1%

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

   9. Taylor expanded in s around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1 \cdot s + \frac{1}{2} \cdot x}{{s}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(-1 \cdot s + \frac{1}{2} \cdot x\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x + -1 \cdot s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x + \left(\mathsf{neg}\left(s\right)\right)\right), \left({s}^{2}\right)\right)\right)\right)\right) \]

       4. unsub-neg N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x - s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{1}{2} \cdot x\right), s\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \left({s}^{2}\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 95.4%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), s\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Simplified 95.4%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5 - s}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/ 1.0 (+ 2.0 (* x (+ (/ -1.0 s) (* x (/ 0.5 (* s s)))))))
   (/ 1.0 (+ 1.0 (/ -1.0 (+ -1.0 (/ (- (/ (* (* x x) -0.5) s) x) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (0.5f / (s * s))))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (((((x * x) * -0.5f) / s) - x) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (0.5e0 / (s * s))))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (((((x * x) * (-0.5e0)) / s) - x) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(0.5) / Float32(s * s)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) + Float32(Float32(Float32(Float32(Float32(x * x) * Float32(-0.5)) / s) - x) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (single(0.5) / (s * s))))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) + (((((x * x) * single(-0.5)) / s) - x) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 84.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 84.2%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in s around -inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + -1 \cdot \frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 + \left(\mathsf{neg}\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)\right)\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(1 - \color{blue}{\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\frac{-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}}{s}\right)}\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(-1 \cdot x + \frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), \color{blue}{s}\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + -1 \cdot x\right), s\right)\right)\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} + \left(\mathsf{neg}\left(x\right)\right)\right), s\right)\right)\right)\right)\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s} - x\right), s\right)\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1}{2} \cdot \frac{{x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\frac{-1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \frac{-1}{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{{x}^{2} \cdot \left(-1 + \frac{1}{2}\right)}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      12. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}}{s}\right), x\right), s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-1 \cdot {x}^{2} + \frac{1}{2} \cdot {x}^{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      14. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \left(-1 + \frac{1}{2}\right)\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f32 95.5%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{2}\right), s\right), x\right), s\right)\right)\right)\right)\right) \]

   7. Simplified 95.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 + \frac{\frac{\left(x \cdot x\right) \cdot -0.5}{s} - x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 90.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14)
   (/ (* (* s (* s s)) -6.0) (* x (* x x)))
   (if (<= x -1.0000000195414814e-25)
     (/ 1.0 (+ 2.0 (* x (/ (* x 0.5) (* s s)))))
     (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = ((s * (s * s)) * -6.0f) / (x * (x * x));
	} else if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((x * 0.5f) / (s * s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = ((s * (s * s)) * (-6.0e0)) / (x * (x * x))
    else if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * 0.5e0) / (s * s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	elseif (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(0.5)) / Float32(s * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = ((s * (s * s)) * single(-6.0)) / (x * (x * x));
	elseif (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((x * single(0.5)) / (s * s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.2%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-6 \cdot {s}^{3}}{\color{blue}{{x}^{3}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-6 \cdot {s}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left({s}^{3} \cdot -6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{3}\right), -6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot \left(s \cdot s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot {s}^{2}\right), -6\right), \left({x}^{3}\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2}\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left({x}^{3}\right)\right) \]

      10. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      14. *-lowering-*.f32 88.2%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right), -6\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   8. Simplified 88.2%

      \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]

   if -1.99999996e-14 < x < -1.00000002e-25

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 81.1%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 99.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2} \cdot x}{\color{blue}{{s}^{2}}}\right)\right)\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f32 91.3%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Simplified 91.3%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 90.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14)
   (/ (* s (* (* s s) -6.0)) (* x (* x x)))
   (if (<= x -1.0000000195414814e-25)
     (/ 1.0 (+ 2.0 (* x (/ (* x 0.5) (* s s)))))
     (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = (s * ((s * s) * -6.0f)) / (x * (x * x));
	} else if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((x * 0.5f) / (s * s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = (s * ((s * s) * (-6.0e0))) / (x * (x * x))
    else if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * 0.5e0) / (s * s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(Float32(s * Float32(Float32(s * s) * Float32(-6.0))) / Float32(x * Float32(x * x)));
	elseif (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(0.5)) / Float32(s * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = (s * ((s * s) * single(-6.0))) / (x * (x * x));
	elseif (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((x * single(0.5)) / (s * s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.2%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} + \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{-1}{6} \cdot \frac{x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{-1}{6} \cdot x}{s \cdot \left(s \cdot s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{s}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(\left(\mathsf{neg}\left(s\right)\right)\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\mathsf{fma}\left(\frac{1}{2}, \frac{1}{s \cdot s}, \mathsf{neg}\left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. fmm-undef N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{1}{2} \cdot \frac{1}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{s \cdot s} - \frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\left(\frac{\frac{1}{2}}{s \cdot s}\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \left(\frac{\frac{\frac{-1}{6} \cdot x}{s \cdot s}}{\mathsf{neg}\left(s\right)}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\left(\frac{\frac{-1}{6} \cdot x}{s \cdot s}\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      18. neg-lowering-neg.f32 87.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, s\right)\right), \mathsf{neg.f32}\left(s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 87.2%

      \[\leadsto \frac{1}{2 + x \cdot \left(x \cdot \color{blue}{\left(\frac{0.5}{s \cdot s} - \frac{\frac{x \cdot -0.16666666666666666}{s \cdot s}}{-s}\right)} + \frac{-1}{s}\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-6 \cdot {s}^{3}}{\color{blue}{{x}^{3}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-6 \cdot {s}^{3}\right), \color{blue}{\left({x}^{3}\right)}\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left({s}^{3} \cdot -6\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot \left(s \cdot s\right)\right) \cdot -6\right), \left({x}^{3}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot {s}^{2}\right) \cdot -6\right), \left({x}^{3}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left({s}^{2} \cdot -6\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left({s}^{2} \cdot -6\right)\right), \left({\color{blue}{x}}^{3}\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left({s}^{2}\right), -6\right)\right), \left({x}^{3}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\left(s \cdot s\right), -6\right)\right), \left({x}^{3}\right)\right) \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \left({x}^{3}\right)\right) \]

      11. cube-mult N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \left(x \cdot {x}^{\color{blue}{2}}\right)\right) \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \mathsf{*.f32}\left(x, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      15. *-lowering-*.f32 87.1%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -6\right)\right), \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right) \]

   10. Simplified 87.1%

       \[\leadsto \color{blue}{\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}} \]

   if -1.99999996e-14 < x < -1.00000002e-25

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 81.1%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 99.1%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 99.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2} \cdot x}{\color{blue}{{s}^{2}}}\right)\right)\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f32 91.3%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Simplified 91.3%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \mathbf{elif}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 90.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/ 1.0 (+ 2.0 (* x (+ (/ -1.0 s) (* x (/ 0.5 (* s s)))))))
   (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (0.5f / (s * s))))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (0.5e0 / (s * s))))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(0.5) / Float32(s * s)))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (single(0.5) / (s * s))))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 84.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 84.2%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 88.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000195414814e-25)
   (/ 1.0 (+ 2.0 (* x (/ (* x 0.5) (* s s)))))
   (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000195414814e-25f) {
		tmp = 1.0f / (2.0f + (x * ((x * 0.5f) / (s * s))));
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000195414814e-25)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * 0.5e0) / (s * s))))
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000195414814e-25))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(0.5)) / Float32(s * s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000195414814e-25))
		tmp = single(1.0) / (single(2.0) + (x * ((x * single(0.5)) / (s * s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000002e-25

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.6%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 84.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 84.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2} \cdot x}{\color{blue}{{s}^{2}}}\right)\right)\right)\right) \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right) \]

       5. *-lowering-*.f32 80.6%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{1}{2}, x\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right) \]

   11. Simplified 80.6%

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

   if -1.00000002e-25 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 93.8%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 93.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000195414814 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot 0.5}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 84.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000031374395e-22)
   (/ (/ (* 2.0 (* s s)) x) x)
   (/ 1.0 (+ 1.0 (/ -1.0 (- -1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000031374395e-22f) {
		tmp = ((2.0f * (s * s)) / x) / x;
	} else {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f - (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000031374395e-22)) then
        tmp = ((2.0e0 * (s * s)) / x) / x
    else
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / ((-1.0e0) - (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000031374395e-22))
		tmp = Float32(Float32(Float32(Float32(2.0) * Float32(s * s)) / x) / x);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(-1.0) / Float32(Float32(-1.0) - Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000031374395e-22))
		tmp = ((single(2.0) * (s * s)) / x) / x;
	else
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / (single(-1.0) - (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000003e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.3%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 83.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 83.0%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \left(2 \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \frac{2 \cdot 1}{x}\right), x\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \frac{2}{x}\right), x\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{/.f32}\left(\left(\frac{{s}^{2} \cdot 2}{x}\right), x\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f32}\left(\left(\frac{2 \cdot {s}^{2}}{x}\right), x\right) \]

       13. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(2 \cdot {s}^{2}\right), x\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left({s}^{2}\right)\right), x\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left(s \cdot s\right)\right), x\right), x\right) \]

       16. *-lowering-*.f32 73.5%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(s, s\right)\right), x\right), x\right) \]

   11. Simplified 73.5%

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

   if -1.00000003e-22 < x

   1. Initial program 99.9%

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

      1. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) + e^{\color{blue}{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

      2. associate--r- N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \color{blue}{\left(-1 - e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)}\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(0 - \left(-1 + \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\left(0 - -1\right) - \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(1 - \left(\mathsf{neg}\left(\color{blue}{e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \color{blue}{\left(\mathsf{neg}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

      8. exp-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\mathsf{neg}\left(\frac{1}{e^{\frac{x}{s}}}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{e^{\frac{x}{s}}}}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \left(\frac{-1}{e^{\color{blue}{\frac{x}{s}}}}\right)\right)\right) \]

      11. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(e^{\frac{x}{s}}\right)}\right)\right)\right) \]

      12. exp-lowering-exp.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      13. /-lowering-/.f32 99.9%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \color{blue}{\left(1 + \frac{x}{s}\right)}\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \left(\frac{x}{s} + \color{blue}{1}\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\left(\frac{x}{s}\right), \color{blue}{1}\right)\right)\right)\right) \]

      3. /-lowering-/.f32 92.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(1, \mathsf{/.f32}\left(-1, \mathsf{+.f32}\left(\mathsf{/.f32}\left(x, s\right), 1\right)\right)\right)\right) \]

   7. Simplified 92.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{-1}{-1 - \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.0% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(s \cdot s\right)}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000031374395e-22) (/ (/ (* 2.0 (* s s)) x) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000031374395e-22f) {
		tmp = ((2.0f * (s * s)) / x) / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000031374395e-22)) then
        tmp = ((2.0e0 * (s * s)) / x) / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000031374395e-22))
		tmp = Float32(Float32(Float32(Float32(2.0) * Float32(s * s)) / x) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000031374395e-22))
		tmp = ((single(2.0) * (s * s)) / x) / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000003e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.3%

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 83.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 83.0%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

       4. *-commutative N/A

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

       5. associate-*r/ N/A

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

       6. metadata-eval N/A

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

       7. associate-*r/ N/A

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

       8. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \left(2 \cdot \frac{1}{x}\right)\right), \color{blue}{x}\right) \]

       9. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \frac{2 \cdot 1}{x}\right), x\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot \frac{2}{x}\right), x\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{/.f32}\left(\left(\frac{{s}^{2} \cdot 2}{x}\right), x\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f32}\left(\left(\frac{2 \cdot {s}^{2}}{x}\right), x\right) \]

       13. /-lowering-/.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(2 \cdot {s}^{2}\right), x\right), x\right) \]

       14. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left({s}^{2}\right)\right), x\right), x\right) \]

       15. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left(s \cdot s\right)\right), x\right), x\right) \]

       16. *-lowering-*.f32 73.5%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(s, s\right)\right), x\right), x\right) \]

   11. Simplified 73.5%

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

   if -1.00000003e-22 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 49.3%

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

Alternative 18: 57.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot -2\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000031374395e-22) (/ (* s (* s -2.0)) (* x x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000031374395e-22f) {
		tmp = (s * (s * -2.0f)) / (x * x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000031374395e-22)) then
        tmp = (s * (s * (-2.0e0))) / (x * x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000031374395e-22))
		tmp = Float32(Float32(s * Float32(s * Float32(-2.0))) / Float32(x * x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000031374395e-22))
		tmp = (s * (s * single(-2.0))) / (x * x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000003e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 42.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 42.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{-1 \cdot s + -2 \cdot \frac{{s}^{2}}{x}}{x} \]

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s - 2 \cdot \frac{{s}^{2}}{x}\right), \color{blue}{x}\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{s}^{2}}{x}\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s + -2 \cdot \frac{{s}^{2}}{x}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} + -1 \cdot s\right), x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} + \left(\mathsf{neg}\left(s\right)\right)\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} - s\right), x\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x}\right), s\right), x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-2 \cdot {s}^{2}}{x}\right), s\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}}{x}\right), s\right), x\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-2 \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({s}^{2} \cdot -2\right), x\right), s\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{2}\right), -2\right), x\right), s\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot s\right), -2\right), x\right), s\right), x\right) \]

      18. *-lowering-*.f32 32.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -2\right), x\right), s\right), x\right) \]

   8. Simplified 32.0%

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

   9. Taylor expanded in s around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot {s}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left({s}^{2} \cdot -2\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\left(s \cdot s\right) \cdot -2\right), \left({x}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left(s \cdot -2\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left(-2 \cdot s\right)\right), \left({x}^{2}\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(-2 \cdot s\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(s \cdot -2\right)\right), \left({x}^{2}\right)\right) \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -2\right)\right), \left({x}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -2\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       11. *-lowering-*.f32 72.8%

           \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, -2\right)\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   11. Simplified 72.8%

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

   if -1.00000003e-22 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 49.3%

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

Alternative 19: 57.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \frac{\frac{s \cdot s}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000031374395e-22) (* -2.0 (/ (/ (* s s) x) x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000031374395e-22f) {
		tmp = -2.0f * (((s * s) / x) / x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.000000031374395e-22)) then
        tmp = (-2.0e0) * (((s * s) / x) / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.000000031374395e-22))
		tmp = Float32(Float32(-2.0) * Float32(Float32(Float32(s * s) / x) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.000000031374395e-22))
		tmp = single(-2.0) * (((s * s) / x) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000003e-22

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 42.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 42.3%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

         \[\leadsto \frac{-1 \cdot s + -2 \cdot \frac{{s}^{2}}{x}}{x} \]

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s - 2 \cdot \frac{{s}^{2}}{x}\right), \color{blue}{x}\right) \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{s}^{2}}{x}\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-1 \cdot s + -2 \cdot \frac{{s}^{2}}{x}\right), x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} + -1 \cdot s\right), x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} + \left(\mathsf{neg}\left(s\right)\right)\right), x\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x} - s\right), x\right) \]

      10. --lowering--.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(-2 \cdot \frac{{s}^{2}}{x}\right), s\right), x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{-2 \cdot {s}^{2}}{x}\right), s\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}}{x}\right), s\right), x\right) \]

      13. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-2 \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({s}^{2} \cdot -2\right), x\right), s\right), x\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({s}^{2}\right), -2\right), x\right), s\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(s \cdot s\right), -2\right), x\right), s\right), x\right) \]

      18. *-lowering-*.f32 32.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), -2\right), x\right), s\right), x\right) \]

   8. Simplified 32.0%

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

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{\left(s \cdot s\right) \cdot -2}{x} + \left(\mathsf{neg}\left(s\right)\right)\right), x\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\frac{s \cdot \left(s \cdot -2\right)}{x} + \left(\mathsf{neg}\left(s\right)\right)\right), x\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \frac{s \cdot -2}{x} + \left(\mathsf{neg}\left(s\right)\right)\right), x\right) \]

      4. neg-mul-1 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \frac{s \cdot -2}{x} + -1 \cdot s\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \frac{s \cdot -2}{x} + s \cdot -1\right), x\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{/.f32}\left(\left(s \cdot \left(\frac{s \cdot -2}{x} + -1\right)\right), x\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \left(\frac{s \cdot -2}{x} + -1\right)\right), x\right) \]

      8. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{s \cdot -2}{x}\right), -1\right)\right), x\right) \]

      9. div-inv N/A

         \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\left(s \cdot -2\right) \cdot \frac{1}{x}\right), -1\right)\right), x\right) \]

      10. associate-*r* N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(s \cdot \left(-2 \cdot \frac{1}{x}\right)\right), -1\right)\right), x\right) \]

      11. div-inv N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(s \cdot \frac{-2}{x}\right), -1\right)\right), x\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(s \cdot \frac{1}{\frac{x}{-2}}\right), -1\right)\right), x\right) \]

      13. un-div-inv N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\left(\frac{s}{\frac{x}{-2}}\right), -1\right)\right), x\right) \]

      14. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(s, \left(\frac{x}{-2}\right)\right), -1\right)\right), x\right) \]

      15. /-lowering-/.f32 32.0%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(s, \mathsf{+.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(x, -2\right)\right), -1\right)\right), x\right) \]

   10. Applied egg-rr 32.0%

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

   11. Taylor expanded in s around inf

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

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \color{blue}{\left(\frac{{s}^{2}}{{x}^{2}}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \left(\frac{{s}^{2}}{x \cdot \color{blue}{x}}\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \left(\frac{\frac{{s}^{2}}{x}}{\color{blue}{x}}\right)\right) \]

       4. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \mathsf{/.f32}\left(\left(\frac{{s}^{2}}{x}\right), \color{blue}{x}\right)\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left({s}^{2}\right), x\right), x\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(s \cdot s\right), x\right), x\right)\right) \]

       7. *-lowering-*.f32 72.7%

          \[\leadsto \mathsf{*.f32}\left(-2, \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(s, s\right), x\right), x\right)\right) \]

   13. Simplified 72.7%

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

   if -1.00000003e-22 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 49.3%

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

Alternative 20: 48.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.9999999593223797e-31) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.9999999593223797e-31) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999996e-31

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 54.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 54.2%

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

   if 1.99999996e-31 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 38.5%

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

Alternative 21: 46.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14) (/ -1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = -1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = (-1.0e0) / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = single(-1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 41.9%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 41.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{s}{x}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 37.4%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{s}{x}\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{neg}\left(\frac{1}{\frac{x}{s}}\right) \]

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

      5. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{x}{s}\right)}\right) \]

      6. /-lowering-/.f32 41.9%

         \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right) \]

   10. Applied egg-rr 41.9%

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

   if -1.99999996e-14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 48.4%

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

Alternative 22: 45.6% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-14) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-14f) {
		tmp = s / -x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-14)) then
        tmp = s / -x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-14))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-14))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 41.9%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 41.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{s}{x}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 37.4%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 37.4%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{s}{x}\right) \]

      2. distribute-frac-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(s\right)}{\color{blue}{x}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(s\right)\right), \color{blue}{x}\right) \]

      4. neg-lowering-neg.f32 37.4%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(s\right), x\right) \]

   10. Applied egg-rr 37.4%

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

   if -1.99999996e-14 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 48.4%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-14}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 34.8% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 35.0%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))