Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.8s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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) * (-0.5e0))) ** 2.0e0))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(((x / s) * single(-0.5))) ^ single(2.0)));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg N/A

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

   2. exp-neg N/A

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

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

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

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

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

   5. /-lowering-/.f32 99.8%

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

4. Applied egg-rr 99.8%

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

   1. inv-pow N/A

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

   2. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-exp N/A

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

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

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

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

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

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

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

   11. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{x}{s} \cdot -0.5}\right)}^{2}}} \]
7. 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 99.8%

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

   1. distribute-frac-neg N/A

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

   2. exp-neg N/A

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

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

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

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

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

   5. /-lowering-/.f32 99.8%

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

4. Applied egg-rr 99.8%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 93.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x + \frac{0.5 \cdot \left(x \cdot x\right) + \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{s}}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/
    1.0
    (+
     1.0
     (/
      1.0
      (+
       1.0
       (/
        (+
         x
         (/ (+ (* 0.5 (* x x)) (/ (* x (* (* x x) 0.16666666666666666)) s)) s))
        s)))))
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (+ (* -0.16666666666666666 (/ x (* s (* s s)))) (/ 0.5 (* s s))))
       (/ -1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + ((x + (((0.5f * (x * x)) + ((x * ((x * x) * 0.16666666666666666f)) / s)) / s)) / s))));
	} else {
		tmp = 1.0f / (2.0f + (x * ((x * ((-0.16666666666666666f * (x / (s * (s * s)))) + (0.5f / (s * s)))) + (-1.0f / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + ((x + (((0.5e0 * (x * x)) + ((x * ((x * x) * 0.16666666666666666e0)) / s)) / s)) / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (x * ((x * (((-0.16666666666666666e0) * (x / (s * (s * s)))) + (0.5e0 / (s * s)))) + ((-1.0e0) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		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(Float32(x * x) * Float32(0.16666666666666666))) / s)) / s)) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(-0.16666666666666666) * Float32(x / Float32(s * Float32(s * s)))) + Float32(Float32(0.5) / Float32(s * s)))) + Float32(Float32(-1.0) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		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))));
	else
		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))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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 94.6%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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 96.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x + \frac{0.5 \cdot \left(x \cdot x\right) + \frac{x \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)}{s}}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (- x (* (* x x) (/ -0.5 s))) s)))))
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (+ (* -0.16666666666666666 (/ x (* s (* s s)))) (/ 0.5 (* s s))))
       (/ -1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + ((x - ((x * x) * (-0.5f / s))) / s))));
	} else {
		tmp = 1.0f / (2.0f + (x * ((x * ((-0.16666666666666666f * (x / (s * (s * s)))) + (0.5f / (s * s)))) + (-1.0f / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + ((x - ((x * x) * ((-0.5e0) / s))) / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (x * ((x * (((-0.16666666666666666e0) * (x / (s * (s * s)))) + (0.5e0 / (s * s)))) + ((-1.0e0) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(x - Float32(Float32(x * x) * Float32(Float32(-0.5) / s))) / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(-0.16666666666666666) * Float32(x / Float32(s * Float32(s * s)))) + Float32(Float32(0.5) / Float32(s * s)))) + Float32(Float32(-1.0) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + ((x - ((x * x) * (single(-0.5) / s))) / s))));
	else
		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))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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

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

   7. Simplified 93.8%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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 96.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(-0.16666666666666666 \cdot \frac{x}{s \cdot \left(s \cdot s\right)} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.4000000059604645)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (- x (* (* x x) (/ -0.5 s))) s)))))
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 s)
       (* x (/ (+ 0.5 (/ (* x -0.16666666666666666) s)) (* s s)))))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.4000000059604645f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + ((x - ((x * x) * (-0.5f / s))) / s))));
	} else {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((0.5f + ((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 / -s) <= 0.4000000059604645e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + ((x - ((x * x) * ((-0.5e0) / s))) / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((0.5e0 + ((x * (-0.16666666666666666e0)) / s)) / (s * s))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(x - Float32(Float32(x * x) * Float32(Float32(-0.5) / s))) / s)))));
	else
		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)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.4000000059604645))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + ((x - ((x * x) * (single(-0.5) / s))) / s))));
	else
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((single(0.5) + ((x * single(-0.16666666666666666)) / s)) / (s * s))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.400000006

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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

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

   7. Simplified 96.1%

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

   if 0.400000006 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 92.6%

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

   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 92.0%

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

   8. Simplified 92.0%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.4000000059604645:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (- x (* (* x x) (/ -0.5 s))) s)))))
   (/ (/ (* -6.0 (/ (* s (* s s)) x)) x) x)))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + ((x - ((x * x) * (-0.5f / s))) / s))));
	} else {
		tmp = ((-6.0f * ((s * (s * s)) / x)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + ((x - ((x * x) * ((-0.5e0) / s))) / s))))
    else
        tmp = (((-6.0e0) * ((s * (s * s)) / x)) / x) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(x - Float32(Float32(x * x) * Float32(Float32(-0.5) / s))) / s)))));
	else
		tmp = Float32(Float32(Float32(Float32(-6.0) * Float32(Float32(s * Float32(s * s)) / x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + ((x - ((x * x) * (single(-0.5) / s))) / s))));
	else
		tmp = ((single(-6.0) * ((s * (s * s)) / x)) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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

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

   7. Simplified 93.8%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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 96.2%

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

   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 77.5%

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

   8. Simplified 77.5%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f32 95.4%

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

   10. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x - \left(x \cdot x\right) \cdot \frac{-0.5}{s}}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5 - \frac{x}{s} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -5.0)
     1.0
     (if (<= t_0 0.5)
       (- 0.5 (* (/ x s) -0.25))
       (/ (* -2.0 (/ (* s s) x)) x)))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 0.5f) {
		tmp = 0.5f - ((x / s) * -0.25f);
	} else {
		tmp = (-2.0f * ((s * s) / x)) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-5.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 0.5e0) then
        tmp = 0.5e0 - ((x / s) * (-0.25e0))
    else
        tmp = ((-2.0e0) * ((s * s) / x)) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(0.5) - Float32(Float32(x / s) * Float32(-0.25)));
	else
		tmp = Float32(Float32(Float32(-2.0) * Float32(Float32(s * s) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = single(1.0);
	elseif (t_0 <= single(0.5))
		tmp = single(0.5) - ((x / s) * single(-0.25));
	else
		tmp = (single(-2.0) * ((s * s) / x)) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \frac{x}{s}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \left(\frac{x}{s} \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(\frac{x}{s}\right), \color{blue}{\frac{-1}{4}}\right)\right) \]

      6. /-lowering-/.f32 96.6%

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{4}\right)\right) \]

   5. Simplified 96.6%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 39.6%

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

   5. Simplified 39.6%

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

   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 36.3%

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

   8. Simplified 36.3%

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

   9. Taylor expanded in s around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f32 84.5%

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

   11. Simplified 84.5%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;0.5 - \frac{x}{s} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5 - \frac{x}{s} \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -5.0)
     1.0
     (if (<= t_0 0.5) (- 0.5 (* (/ x s) -0.25)) (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 0.5f) {
		tmp = 0.5f - ((x / s) * -0.25f);
	} else {
		tmp = -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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-5.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 0.5e0) then
        tmp = 0.5e0 - ((x / s) * (-0.25e0))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(Float32(0.5) - Float32(Float32(x / s) * Float32(-0.25)));
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = single(1.0);
	elseif (t_0 <= single(0.5))
		tmp = single(0.5) - ((x / s) * single(-0.25));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. cancel-sign-sub-inv N/A

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

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

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{4} \cdot \frac{x}{s}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \left(\frac{x}{s} \cdot \color{blue}{\frac{-1}{4}}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\left(\frac{x}{s}\right), \color{blue}{\frac{-1}{4}}\right)\right) \]

      6. /-lowering-/.f32 96.6%

         \[\leadsto \mathsf{\_.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{4}\right)\right) \]

   5. Simplified 96.6%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 39.6%

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

   5. Simplified 39.6%

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

   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 36.3%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   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 39.6%

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

   10. Applied egg-rr 39.6%

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

4. Final simplification 75.4%

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

Alternative 10: 74.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -5.0) 1.0 (if (<= t_0 0.5) 0.5 (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 0.5f) {
		tmp = 0.5f;
	} else {
		tmp = -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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-5.0e0)) then
        tmp = 1.0e0
    else if (t_0 <= 0.5e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(1.0);
	elseif (t_0 <= Float32(0.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = single(1.0);
	elseif (t_0 <= single(0.5))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s) < 0.5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 87.2%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 39.6%

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

   5. Simplified 39.6%

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

   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 36.3%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 36.3%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   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 39.6%

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

   10. Applied egg-rr 39.6%

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

4. Final simplification 73.1%

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

Alternative 11: 92.4% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ (/ (* -6.0 (/ (* s (* s s)) x)) x) x)))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = ((-6.0f * ((s * (s * s)) / x)) / x) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = (((-6.0e0) * ((s * (s * s)) / x)) / x) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(Float32(-6.0) * Float32(Float32(s * Float32(s * s)) / x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = ((single(-6.0) * ((s * (s * s)) / x)) / x) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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 92.4%

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

   7. Simplified 92.4%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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 96.2%

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

   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 77.5%

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

   8. Simplified 77.5%

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f32 95.4%

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

   10. Applied egg-rr 95.4%

       \[\leadsto \color{blue}{\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-6 \cdot \frac{s \cdot \left(s \cdot s\right)}{x}}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 88.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 50.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ (* -2.0 (/ (* s s) x)) x)))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 50.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = (-2.0f * ((s * s) / x)) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 50.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = ((-2.0e0) * ((s * s) / x)) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(Float32(-2.0) * Float32(Float32(s * s) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(50.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = (single(-2.0) * ((s * s) / x)) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.8%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 99.8%

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

   4. Applied egg-rr 99.8%

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

   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 92.4%

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

   7. Simplified 92.4%

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

   if 50 < (/.f32 (neg.f32 x) s)

   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 40.4%

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

   5. Simplified 40.4%

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

   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 37.0%

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

   8. Simplified 37.0%

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

   9. Taylor expanded in s around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f32 87.8%

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

   11. Simplified 87.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 50:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{s \cdot 2 - x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -5.0) 1.0 (/ 1.0 (/ (- (* s 2.0) x) s))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -5.0f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f / (((s * 2.0f) - x) / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-5.0e0)) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 / (((s * 2.0e0) - x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-5.0))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(s * Float32(2.0)) - x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-5.0))
		tmp = single(1.0);
	else
		tmp = single(1.0) / (((s * single(2.0)) - x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in s around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f32 60.4%

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

   8. Simplified 60.4%

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

4. Final simplification 74.7%

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

Alternative 14: 76.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -5.0) 1.0 (/ 1.0 (+ 2.0 (/ -1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -5.0f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f / (2.0f + (-1.0f / (s / x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-5.0e0)) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 / (2.0e0 + ((-1.0e0) / (s / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-5.0))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(-1.0) / Float32(s / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-5.0))
		tmp = single(1.0);
	else
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 60.4%

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

   5. Simplified 60.4%

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

      1. clear-num N/A

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

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

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

      3. /-lowering-/.f32 60.4%

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

   7. Applied egg-rr 60.4%

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

4. Final simplification 74.7%

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

Alternative 15: 76.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -5.0) 1.0 (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -5.0f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f / (2.0f + (x * (-1.0f / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-5.0e0)) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 / (2.0e0 + (x * ((-1.0e0) / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-5.0))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(-1.0) / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-5.0))
		tmp = single(1.0);
	else
		tmp = single(1.0) / (single(2.0) + (x * (single(-1.0) / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 60.4%

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

   5. Simplified 60.4%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. /-lowering-/.f32 60.4%

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

   7. Applied egg-rr 60.4%

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

4. Final simplification 74.7%

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

Alternative 16: 76.1% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -5.0) 1.0 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -5.0f) {
		tmp = 1.0f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-5.0e0)) then
        tmp = 1.0e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-5.0))
		tmp = Float32(1.0);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-5.0))
		tmp = single(1.0);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 99.5%

       \[\leadsto \color{blue}{1} \]

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 60.4%

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

   5. Simplified 60.4%

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

4. Final simplification 74.7%

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

Alternative 17: 69.2% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0 - \frac{s}{x}\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999873689376e-5)
   (- 0.0 (/ s x))
   (if (<= x 1.999999936531045e-20) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999873689376e-5f) {
		tmp = 0.0f - (s / x);
	} else if (x <= 1.999999936531045e-20f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999873689376e-5)) then
        tmp = 0.0e0 - (s / x)
    else if (x <= 1.999999936531045e-20) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999873689376e-5))
		tmp = Float32(Float32(0.0) - Float32(s / x));
	elseif (x <= Float32(1.999999936531045e-20))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999873689376e-5))
		tmp = single(0.0) - (s / x);
	elseif (x <= single(1.999999936531045e-20))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4.99999987e-5

   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 53.3%

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

   5. Simplified 53.3%

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

   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 48.5%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 48.5%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{s}{x}\right) \]

      2. distribute-neg-frac 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 48.5%

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

   10. Applied egg-rr 48.5%

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

   if -4.99999987e-5 < x < 1.99999994e-20

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 58.0%

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

   if 1.99999994e-20 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 94.0%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0 - \frac{s}{x}\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.0% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 1.999999936531045e-20) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.999999936531045e-20f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.999999936531045e-20) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.999999936531045e-20))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.999999936531045e-20))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.99999994e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 35.8%

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

   if 1.99999994e-20 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

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

      2. exp-neg N/A

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

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

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

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

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

      5. /-lowering-/.f32 100.0%

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

   4. Applied egg-rr 100.0%

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

   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

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

   7. Simplified 97.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   9. Step-by-step derivation

   10. Simplified 94.0%

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

Alternative 19: 34.6% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 34.1%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))