Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 14

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, 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 56.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.00000011871114e-34)
   (/ (* s 0.5) s)
   (if (<= (- x) 39999999311872.0)
     (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ (/ x s) 2.0)))
     (* s (/ -1.0 (* s (- (/ x s) 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.00000011871114e-34f) {
		tmp = (s * 0.5f) / s;
	} else if (-x <= 39999999311872.0f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / ((x / s) + 2.0f));
	} else {
		tmp = s * (-1.0f / (s * ((x / s) - 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-5.00000011871114e-34)) then
        tmp = (s * 0.5e0) / s
    else if (-x <= 39999999311872.0e0) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / ((x / s) + 2.0e0))
    else
        tmp = s * ((-1.0e0) / (s * ((x / s) - 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-5.00000011871114e-34))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	elseif (Float32(-x) <= Float32(39999999311872.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(s * Float32(Float32(-1.0) / Float32(s * Float32(Float32(x / s) - Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-5.00000011871114e-34))
		tmp = (s * single(0.5)) / s;
	elseif (-x <= single(39999999311872.0))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / ((x / s) + single(2.0)));
	else
		tmp = s * (single(-1.0) / (s * ((x / s) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -5.00000012e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 19.8%

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

      1. associate-*r/ 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in s around 0 19.8%

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

   7. Taylor expanded in x around 0 34.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   if -5.00000012e-34 < (neg.f32 x) < 3.99999993e13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.9%

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

      1. neg-mul-1 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

      1. sub-neg 45.9%

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

      2. flip-+ 61.2%

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

      3. metadata-eval 61.2%

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

      4. distribute-neg-frac2 61.2%

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

      5. distribute-neg-frac2 61.2%

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

      6. distribute-neg-frac2 61.2%

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

   7. Applied egg-rr 61.2%

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

      1. clear-num 61.2%

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

      2. frac-times 64.2%

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

      3. *-un-lft-identity 64.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 66.7%

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

      6. sqr-neg 66.7%

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

      7. sqrt-unprod 63.8%

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

      8. add-sqr-sqrt 63.8%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 64.2%

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

      13. add-sqr-sqrt 64.2%

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

   9. Applied egg-rr 64.2%

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

       1. *-un-lft-identity 64.2%

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

       2. times-frac 61.2%

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

       3. clear-num 61.2%

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

       4. frac-times 61.6%

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

       5. sqr-neg 61.6%

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

       6. frac-times 61.2%

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

       7. associate-*l/ 61.2%

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

       8. *-un-lft-identity 61.2%

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

       9. times-frac 70.2%

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

       10. add-sqr-sqrt -0.0%

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

       11. sqrt-unprod 66.7%

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

       12. sqr-neg 66.7%

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

       13. sqrt-unprod 69.9%

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

       14. add-sqr-sqrt 69.9%

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

       15. add-sqr-sqrt -0.0%

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

       16. sqrt-unprod 67.0%

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

       17. sqr-neg 67.0%

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

       18. sqrt-unprod 70.2%

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

       19. add-sqr-sqrt 70.2%

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

   11. Applied egg-rr 70.2%

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

   if 3.99999993e13 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. unsub-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in s around 0 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

      1. associate-/r/ 70.2%

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

   10. Applied egg-rr 70.2%

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

   11. Taylor expanded in s around inf 76.9%

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

       1. neg-mul-1 76.9%

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

       2. sub-neg 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s \cdot \frac{s}{x}} - 4}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.00000011871114e-34)
   (/ (* s 0.5) s)
   (if (<= (- x) 39999999311872.0)
     (/ -1.0 (/ (- (/ x (* s (/ s x))) 4.0) (+ (/ x s) 2.0)))
     (* s (/ -1.0 (* s (- (/ x s) 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.00000011871114e-34f) {
		tmp = (s * 0.5f) / s;
	} else if (-x <= 39999999311872.0f) {
		tmp = -1.0f / (((x / (s * (s / x))) - 4.0f) / ((x / s) + 2.0f));
	} else {
		tmp = s * (-1.0f / (s * ((x / s) - 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-5.00000011871114e-34)) then
        tmp = (s * 0.5e0) / s
    else if (-x <= 39999999311872.0e0) then
        tmp = (-1.0e0) / (((x / (s * (s / x))) - 4.0e0) / ((x / s) + 2.0e0))
    else
        tmp = s * ((-1.0e0) / (s * ((x / s) - 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-5.00000011871114e-34))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	elseif (Float32(-x) <= Float32(39999999311872.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(x / Float32(s * Float32(s / x))) - Float32(4.0)) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(s * Float32(Float32(-1.0) / Float32(s * Float32(Float32(x / s) - Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-5.00000011871114e-34))
		tmp = (s * single(0.5)) / s;
	elseif (-x <= single(39999999311872.0))
		tmp = single(-1.0) / (((x / (s * (s / x))) - single(4.0)) / ((x / s) + single(2.0)));
	else
		tmp = s * (single(-1.0) / (s * ((x / s) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -5.00000012e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 19.8%

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

      1. associate-*r/ 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in s around 0 19.8%

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

   7. Taylor expanded in x around 0 34.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   if -5.00000012e-34 < (neg.f32 x) < 3.99999993e13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.9%

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

      1. neg-mul-1 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

      1. sub-neg 45.9%

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

      2. flip-+ 61.2%

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

      3. metadata-eval 61.2%

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

      4. distribute-neg-frac2 61.2%

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

      5. distribute-neg-frac2 61.2%

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

      6. distribute-neg-frac2 61.2%

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

   7. Applied egg-rr 61.2%

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

      1. clear-num 61.2%

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

      2. frac-times 64.2%

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

      3. *-un-lft-identity 64.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 66.7%

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

      6. sqr-neg 66.7%

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

      7. sqrt-unprod 63.8%

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

      8. add-sqr-sqrt 63.8%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 67.0%

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

      11. sqr-neg 67.0%

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

      12. sqrt-unprod 64.2%

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

      13. add-sqr-sqrt 64.2%

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

   9. Applied egg-rr 64.2%

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

   10. Taylor expanded in x around 0 64.2%

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

   if 3.99999993e13 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. unsub-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in s around 0 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

      1. associate-/r/ 70.2%

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

   10. Applied egg-rr 70.2%

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

   11. Taylor expanded in s around inf 76.9%

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

       1. neg-mul-1 76.9%

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

       2. sub-neg 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s \cdot \frac{s}{x}} - 4}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.9999999880125916e-12)
   (/ (* s 0.5) s)
   (if (<= (- x) 39999999311872.0)
     (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
     (* s (/ -1.0 (* s (- (/ x s) 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.9999999880125916e-12f) {
		tmp = (s * 0.5f) / s;
	} else if (-x <= 39999999311872.0f) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} else {
		tmp = s * (-1.0f / (s * ((x / s) - 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 2.9999999880125916e-12) then
        tmp = (s * 0.5e0) / s
    else if (-x <= 39999999311872.0e0) then
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (x / s))
    else
        tmp = s * ((-1.0e0) / (s * ((x / s) - 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.9999999880125916e-12))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	elseif (Float32(-x) <= Float32(39999999311872.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(x / s)));
	else
		tmp = Float32(s * Float32(Float32(-1.0) / Float32(s * Float32(Float32(x / s) - Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.9999999880125916e-12))
		tmp = (s * single(0.5)) / s;
	elseif (-x <= single(39999999311872.0))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (x / s));
	else
		tmp = s * (single(-1.0) / (s * ((x / s) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 2.99999999e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 34.0%

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

      1. associate-*r/ 34.0%

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

   5. Simplified 34.0%

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

   6. Taylor expanded in s around 0 33.9%

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

   7. Taylor expanded in x around 0 43.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 43.4%

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

   9. Simplified 43.4%

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

   if 2.99999999e-12 < (neg.f32 x) < 3.99999993e13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 14.6%

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

      1. neg-mul-1 14.6%

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

      2. unsub-neg 14.6%

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

   5. Simplified 14.6%

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

      1. sub-neg 14.6%

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

      2. flip-+ 49.2%

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

      3. metadata-eval 49.2%

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

      4. distribute-neg-frac2 49.2%

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

      5. distribute-neg-frac2 49.2%

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

      6. distribute-neg-frac2 49.2%

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

   7. Applied egg-rr 49.2%

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

   8. Taylor expanded in x around inf 47.5%

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

   if 3.99999993e13 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. unsub-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in s around 0 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

      1. associate-/r/ 70.2%

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

   10. Applied egg-rr 70.2%

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

   11. Taylor expanded in s around inf 76.9%

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

       1. neg-mul-1 76.9%

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

       2. sub-neg 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.9999999880125916e-12)
   (/ (* s 0.5) s)
   (if (<= (- x) 39999999311872.0)
     (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (/ x s)))
     (* s (/ -1.0 (* s (- (/ x s) 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.9999999880125916e-12f) {
		tmp = (s * 0.5f) / s;
	} else if (-x <= 39999999311872.0f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (x / s));
	} else {
		tmp = s * (-1.0f / (s * ((x / s) - 2.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 2.9999999880125916e-12) then
        tmp = (s * 0.5e0) / s
    else if (-x <= 39999999311872.0e0) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (x / s))
    else
        tmp = s * ((-1.0e0) / (s * ((x / s) - 2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.9999999880125916e-12))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	elseif (Float32(-x) <= Float32(39999999311872.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(x / s)));
	else
		tmp = Float32(s * Float32(Float32(-1.0) / Float32(s * Float32(Float32(x / s) - Float32(2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.9999999880125916e-12))
		tmp = (s * single(0.5)) / s;
	elseif (-x <= single(39999999311872.0))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (x / s));
	else
		tmp = s * (single(-1.0) / (s * ((x / s) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 2.99999999e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 34.0%

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

      1. associate-*r/ 34.0%

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

   5. Simplified 34.0%

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

   6. Taylor expanded in s around 0 33.9%

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

   7. Taylor expanded in x around 0 43.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 43.4%

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

   9. Simplified 43.4%

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

   if 2.99999999e-12 < (neg.f32 x) < 3.99999993e13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 14.6%

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

      1. neg-mul-1 14.6%

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

      2. unsub-neg 14.6%

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

   5. Simplified 14.6%

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

      1. sub-neg 14.6%

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

      2. flip-+ 49.2%

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

      3. metadata-eval 49.2%

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

      4. distribute-neg-frac2 49.2%

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

      5. distribute-neg-frac2 49.2%

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

      6. distribute-neg-frac2 49.2%

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

   7. Applied egg-rr 49.2%

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

      1. clear-num 49.2%

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

      2. frac-times 49.2%

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

      3. *-un-lft-identity 49.2%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 53.6%

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

      6. sqr-neg 53.6%

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

      7. sqrt-unprod 48.4%

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

      8. add-sqr-sqrt 48.4%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 54.4%

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

      11. sqr-neg 54.4%

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

      12. sqrt-unprod 49.2%

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

      13. add-sqr-sqrt 49.2%

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

   9. Applied egg-rr 49.2%

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

   10. Taylor expanded in x around inf 47.5%

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

   if 3.99999993e13 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 76.9%

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

      1. neg-mul-1 76.9%

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

      2. unsub-neg 76.9%

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

   5. Simplified 76.9%

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

   6. Taylor expanded in s around 0 76.9%

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

      1. *-commutative 76.9%

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

   8. Simplified 76.9%

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

      1. associate-/r/ 70.2%

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

   10. Applied egg-rr 70.2%

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

   11. Taylor expanded in s around inf 76.9%

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

       1. neg-mul-1 76.9%

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

       2. sub-neg 76.9%

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

   13. Simplified 76.9%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{elif}\;-x \leq 39999999311872:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;s \cdot \frac{-1}{s \cdot \left(\frac{x}{s} - 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.00000011871114e-34)
   (/ (* s 0.5) s)
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.00000011871114e-34f) {
		tmp = (s * 0.5f) / s;
	} 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 <= (-5.00000011871114e-34)) then
        tmp = (s * 0.5e0) / s
    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(-5.00000011871114e-34))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	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 <= single(-5.00000011871114e-34))
		tmp = (s * single(0.5)) / s;
	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 (neg.f32 x) < -5.00000012e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 19.8%

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

      1. associate-*r/ 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in s around 0 19.8%

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

   7. Taylor expanded in x around 0 34.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   if -5.00000012e-34 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 55.3%

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

      1. neg-mul-1 55.3%

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

      2. unsub-neg 55.3%

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

   5. Simplified 55.3%

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

      1. clear-num 55.3%

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

      2. associate-/r/ 55.3%

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

   7. Applied egg-rr 55.3%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.00000011871114e-34) (/ (* s 0.5) s) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.00000011871114e-34f) {
		tmp = (s * 0.5f) / s;
	} 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 <= (-5.00000011871114e-34)) then
        tmp = (s * 0.5e0) / s
    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(-5.00000011871114e-34))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	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 <= single(-5.00000011871114e-34))
		tmp = (s * single(0.5)) / s;
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -5.00000012e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 19.8%

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

      1. associate-*r/ 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in s around 0 19.8%

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

   7. Taylor expanded in x around 0 34.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 34.9%

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

   9. Simplified 34.9%

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

   if -5.00000012e-34 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 55.3%

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

      1. neg-mul-1 55.3%

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

      2. unsub-neg 55.3%

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

   5. Simplified 55.3%

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

Alternative 8: 47.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.9999999880125916e-12) (/ (* s 0.5) s) (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.9999999880125916e-12f) {
		tmp = (s * 0.5f) / s;
	} 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) :: tmp
    if (-x <= 2.9999999880125916e-12) then
        tmp = (s * 0.5e0) / s
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.9999999880125916e-12))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.9999999880125916e-12))
		tmp = (s * single(0.5)) / s;
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 2.99999999e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 34.0%

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

      1. associate-*r/ 34.0%

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

   5. Simplified 34.0%

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

   6. Taylor expanded in s around 0 33.9%

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

   7. Taylor expanded in x around 0 43.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 43.4%

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

   9. Simplified 43.4%

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

   if 2.99999999e-12 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 47.8%

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

      1. neg-mul-1 47.8%

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

      2. unsub-neg 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in x around inf 47.3%

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

      1. mul-1-neg 47.3%

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

      2. distribute-frac-neg2 47.3%

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

   8. Simplified 47.3%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 0.00019999999494757503:\\ \;\;\;\;\frac{s \cdot 0.5}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 0.00019999999494757503) (/ (* s 0.5) s) (/ 1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 0.00019999999494757503f) {
		tmp = (s * 0.5f) / s;
	} 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) :: tmp
    if (-x <= 0.00019999999494757503e0) then
        tmp = (s * 0.5e0) / s
    else
        tmp = 1.0e0 / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(s * Float32(0.5)) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(0.00019999999494757503))
		tmp = (s * single(0.5)) / s;
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.99999995e-4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 32.4%

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

      1. associate-*r/ 32.4%

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

   5. Simplified 32.4%

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

   6. Taylor expanded in s around 0 32.3%

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

   7. Taylor expanded in x around 0 41.2%

      \[\leadsto \frac{\color{blue}{0.5 \cdot s}}{s} \]
   8. Step-by-step derivation

      1. *-commutative 41.2%

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

   9. Simplified 41.2%

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

   if 1.99999995e-4 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. neg-mul-1 54.9%

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

      2. unsub-neg 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. neg-mul-1 50.1%

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

   8. Simplified 50.1%

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

      1. clear-num 54.9%

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

      2. inv-pow 54.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 64.3%

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

      5. sqr-neg 64.3%

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

      6. sqrt-unprod 54.9%

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

      7. add-sqr-sqrt 54.9%

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

   10. Applied egg-rr 54.9%

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

       1. unpow-1 54.9%

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

   12. Simplified 54.9%

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

Alternative 10: 47.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0005000000237487257) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0005000000237487257f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0005000000237487257e0)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0005000000237487257))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0005000000237487257))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. neg-mul-1 54.9%

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

      2. unsub-neg 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. neg-mul-1 50.1%

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

   8. Simplified 50.1%

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

      1. clear-num 54.9%

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

      2. inv-pow 54.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 64.3%

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

      5. sqr-neg 64.3%

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

      6. sqrt-unprod 54.9%

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

      7. add-sqr-sqrt 54.9%

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

   10. Applied egg-rr 54.9%

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

       1. unpow-1 54.9%

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

   12. Simplified 54.9%

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

   if -5.00000024e-4 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 41.2%

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

Alternative 11: 45.9% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.9999999880125916e-12) (* s (/ -1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.9999999880125916e-12f) {
		tmp = s * (-1.0f / x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.9999999880125916e-12)) then
        tmp = s * ((-1.0e0) / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.9999999880125916e-12))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.9999999880125916e-12))
		tmp = s * (single(-1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 47.8%

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

      1. neg-mul-1 47.8%

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

      2. unsub-neg 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in s around 0 47.8%

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

      1. *-commutative 47.8%

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

   8. Simplified 47.8%

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

      1. associate-/r/ 43.9%

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

   10. Applied egg-rr 43.9%

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

   11. Taylor expanded in s around 0 43.3%

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

   if -2.99999999e-12 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.9% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.9999999880125916e-12) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.9999999880125916e-12f) {
		tmp = s / -x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.9999999880125916e-12)) then
        tmp = s / -x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.9999999880125916e-12))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.9999999880125916e-12))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.99999999e-12

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 47.8%

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

      1. neg-mul-1 47.8%

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

      2. unsub-neg 47.8%

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

   5. Simplified 47.8%

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

   6. Taylor expanded in x around inf 43.3%

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

      1. associate-*r/ 43.3%

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

      2. neg-mul-1 43.3%

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

   8. Simplified 43.3%

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

   if -2.99999999e-12 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 43.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9999999880125916 \cdot 10^{-12}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0005000000237487257:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0005000000237487257) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0005000000237487257f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0005000000237487257e0)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0005000000237487257))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0005000000237487257))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000024e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.9%

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

      1. neg-mul-1 54.9%

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

      2. unsub-neg 54.9%

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

   5. Simplified 54.9%

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

   6. Taylor expanded in x around inf 50.1%

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

      1. associate-*r/ 50.1%

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

      2. neg-mul-1 50.1%

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

   8. Simplified 50.1%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 61.3%

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

      3. sqr-neg 61.3%

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

      4. sqrt-unprod 50.1%

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

      5. add-sqr-sqrt 50.1%

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

      6. *-un-lft-identity 50.1%

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

   10. Applied egg-rr 50.1%

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

       1. *-lft-identity 50.1%

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

   12. Simplified 50.1%

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

   if -5.00000024e-4 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 41.2%

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

Alternative 14: 35.1% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 32.8%

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

Reproduce

herbie shell --seed 2024151 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))