Logistic function

Percentage Accurate: 99.7% → 99.7%

Time: 5.9s

Alternatives: 9

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: 99.7% 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: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp((-2.0e0)) ** (0.5e0 * (x / s))) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. rec-exp N/A

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

   4. neg-mul-1 N/A

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

   5. pow-exp N/A

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

   6. lift-exp.f32 N/A

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

   7. sqr-pow N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-exp.f32 N/A

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

   12. prod-exp N/A

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

   13. metadata-eval N/A

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

   14. lower-exp.f32 N/A

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{1}{{\left(e^{-2}\right)}^{\left(0.5 \cdot \frac{x}{s}\right)} + 1} \]
8. Add Preprocessing

Alternative 2: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 10000:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 10000.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (- (/ 0.5 (* s s)) (/ (- (/ 1.0 s) (/ 2.0 x)) x)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 10000.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f / (s * s)) - (((1.0f / s) - (2.0f / x)) / 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 (exp((-x / s)) <= 10000.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((((0.5e0 / (s * s)) - (((1.0e0 / s) - (2.0e0 / x)) / x)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(10000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(Float32(1.0) / s) - Float32(Float32(2.0) / x)) / x)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(10000.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((((single(0.5) / (s * s)) - (((single(1.0) / s) - (single(2.0) / x)) / x)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 1e4

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 94.5

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

   7. Applied rewrites 94.5%

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

   if 1e4 < (exp.f32 (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 86.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 10000:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5199999809265137:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (/ (- x) s)) 1.0)) 0.5199999809265137)
   (/ 1.0 (- 2.0 (/ x s)))
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf((-x / s)) + 1.0f)) <= 0.5199999809265137f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 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 ((1.0e0 / (exp((-x / s)) + 1.0e0)) <= 0.5199999809265137e0) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(Float32(-x) / s)) + Float32(1.0))) <= Float32(0.5199999809265137))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) / (exp((-x / s)) + single(1.0))) <= single(0.5199999809265137))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.519999981

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.519999981 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 95.5

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

   7. Applied rewrites 95.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{\frac{-x}{s}} + 1} \leq 0.5199999809265137:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 5: 89.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{-1}{s \cdot s} \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right) - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 5.000000156871975e-23)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (- (- 2.0 (* (/ -1.0 (* s s)) (* (* x x) 0.5))) (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 5.000000156871975e-23f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((2.0f - ((-1.0f / (s * s)) * ((x * x) * 0.5f))) - (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 <= 5.000000156871975e-23) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((2.0e0 - (((-1.0e0) / (s * s)) * ((x * x) * 0.5e0))) - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(5.000000156871975e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(Float32(Float32(-1.0) / Float32(s * s)) * Float32(Float32(x * x) * Float32(0.5)))) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(5.000000156871975e-23))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((single(2.0) - ((single(-1.0) / (s * s)) * ((x * x) * single(0.5)))) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 5.00000016e-23

   1. Initial program 99.7%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 93.2

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

   7. Applied rewrites 93.2%

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

   if 5.00000016e-23 < (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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 14.3%

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

   7. Applied rewrites 74.5%

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

   9. Applied rewrites 86.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 5.000000156871975 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{-1}{s \cdot s} \cdot \left(\left(x \cdot x\right) \cdot 0.5\right)\right) - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.05000000074505806)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (/ (/ x s) s) x) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.05000000074505806f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((((x / s) / s) * x) * 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 / s) <= 0.05000000074505806e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((((x / s) / s) * x) * 0.5e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x / s) / s) * x) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.05000000074505806))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / ((((x / s) / s) * x) * single(0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0500000007

   1. Initial program 99.9%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 96.5

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

   7. Applied rewrites 96.5%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 7.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 78.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{x}{s}}{s} \cdot x\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\left(\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s} + 2\right) - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999841327613e-22)
   (/ 1.0 (- (+ (/ (* (* x x) 0.5) (* s s)) 2.0) (/ x s)))
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999841327613e-22f) {
		tmp = 1.0f / (((((x * x) * 0.5f) / (s * s)) + 2.0f) - (x / s));
	} else {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 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 <= (-4.999999841327613e-22)) then
        tmp = 1.0e0 / (((((x * x) * 0.5e0) / (s * s)) + 2.0e0) - (x / s))
    else
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999841327613e-22))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(x * x) * Float32(0.5)) / Float32(s * s)) + Float32(2.0)) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999841327613e-22))
		tmp = single(1.0) / (((((x * x) * single(0.5)) / (s * s)) + single(2.0)) - (x / s));
	else
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.9999998e-22

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 11.8%

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

   7. Applied rewrites 73.8%

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

   9. Applied rewrites 83.9%

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

   if -4.9999998e-22 < x

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 93.3

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

   7. Applied rewrites 93.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999841327613 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\left(\frac{\left(x \cdot x\right) \cdot 0.5}{s \cdot s} + 2\right) - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-2.0))
		tmp = Float32(0.5);
	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(-2.0))
		tmp = single(0.5);
	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) < -2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 63.8

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

   5. Applied rewrites 63.8%

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

Alternative 9: 35.3% accurate, 128.0× 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 99.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 34.4%

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))