Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 11.1s

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: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. *-commutative N/A

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

   5. log-pow N/A

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

   6. inv-pow N/A

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

   7. lift-/.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lift-/.f32 N/A

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

   10. log-rec N/A

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

   11. lower-neg.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-log1p.f32 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

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

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

   2. lift-/.f32 N/A

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

   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. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

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

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

   2. lift-/.f32 N/A

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

   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. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((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 / ((1.0e0 / exp((x / s))) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

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

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

   2. lift-/.f32 N/A

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

   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. Final simplification 99.8%

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

Alternative 5: 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

Derivation

1. Initial program 99.8%

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

Alternative 6: 66.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.6000000238418579:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{\left(s \cdot s\right) \cdot s}, -0.16666666666666666, \frac{0.5}{s \cdot s}\right), x, \frac{-1}{s}\right), x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.6000000238418579)
   0.5
   (/
    1.0
    (fma
     (fma
      (fma (/ x (* (* s s) s)) -0.16666666666666666 (/ 0.5 (* s s)))
      x
      (/ -1.0 s))
     x
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.6000000238418579f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(fmaf(fmaf((x / ((s * s) * s)), -0.16666666666666666f, (0.5f / (s * s))), x, (-1.0f / s)), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.6000000238418579))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(fma(fma(Float32(x / Float32(Float32(s * s) * s)), Float32(-0.16666666666666666), Float32(Float32(0.5) / Float32(s * s))), x, Float32(Float32(-1.0) / s)), x, Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.600000024

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 51.4%

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

   if 0.600000024 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. lift-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\left(0 - x\right)} \cdot \frac{1}{s}}} \]

      5. flip3-- N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{1}{s}}} \]

      6. frac-times N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{\left({0}^{3} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\left(\color{blue}{0} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      8. sub0-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      9. cube-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3} \cdot \color{blue}{{1}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 1\right)}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      14. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      15. cube-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      16. sub0-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{0 - {x}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{0}^{3}} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      18. lower-/.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   7. Applied rewrites 86.6%

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

Alternative 7: 65.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.6000000238418579:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.6000000238418579)
   0.5
   (/
    1.0
    (fma
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     x
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.6000000238418579f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), x, 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.6000000238418579))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), x, Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.600000024

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 51.4%

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

   if 0.600000024 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 85.1%

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

Alternative 8: 64.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right) \cdot x\right) \cdot \frac{x}{s} - x}{s} + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10000.0)
   0.5
   (/
    1.0
    (+
     (/ (- (* (* (fma -0.16666666666666666 (/ x s) 0.5) x) (/ x s)) x) s)
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((((fmaf(-0.16666666666666666f, (x / s), 0.5f) * x) * (x / s)) - x) / s) + 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)) * x) * Float32(x / s)) - x) / s) + Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

      1. lift-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\left(0 - x\right)} \cdot \frac{1}{s}}} \]

      5. flip3-- N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{1}{s}}} \]

      6. frac-times N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{\left({0}^{3} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\left(\color{blue}{0} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      8. sub0-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      9. cube-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3} \cdot \color{blue}{{1}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 1\right)}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      14. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      15. cube-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      16. sub0-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{0 - {x}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{0}^{3}} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      18. lower-/.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

   4. Applied rewrites 47.2%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 86.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(\mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right) \cdot x\right) \cdot \frac{x}{s} - x}{s} + 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 250:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666}{\left(s \cdot s\right) \cdot s} \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 250.0)
   0.5
   (/ 1.0 (* (/ -0.16666666666666666 (* (* s s) s)) (* (* x x) x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 250.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((-0.16666666666666666f / ((s * s) * s)) * ((x * x) * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 250.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((-0.16666666666666666e0) / ((s * s) * s)) * ((x * x) * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(250.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-0.16666666666666666) / Float32(Float32(s * s) * s)) * Float32(Float32(x * x) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(250.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(-0.16666666666666666) / ((s * s) * s)) * ((x * x) * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 250

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 49.3%

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

   if 250 < (/.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-/.f32 N/A

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

      2. div-inv N/A

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

      3. lift-neg.f32 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\left(0 - x\right)} \cdot \frac{1}{s}}} \]

      5. flip3-- N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}} \cdot \frac{1}{s}}} \]

      6. frac-times N/A

         \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{\left({0}^{3} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\left(\color{blue}{0} - {x}^{3}\right) \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      8. sub0-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\left(\mathsf{neg}\left({x}^{3}\right)\right)} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      9. cube-neg N/A

         \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      10. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3} \cdot 1}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\left(\mathsf{neg}\left(x\right)\right)}^{3} \cdot \color{blue}{{1}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      12. unpow-prod-down N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot 1\right)}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      13. *-rgt-identity N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      14. lift-neg.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\frac{{\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      15. cube-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{\mathsf{neg}\left({x}^{3}\right)}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      16. sub0-neg N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{0 - {x}^{3}}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{1}{1 + e^{\frac{\color{blue}{{0}^{3}} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}} \]

      18. lower-/.f32 N/A

          \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{{0}^{3} - {x}^{3}}{\left(0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right) \cdot s}}}} \]

   4. Applied rewrites 63.0%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-1}{6} \cdot \color{blue}{\frac{{x}^{3}}{{s}^{3}}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 92.1%

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

Alternative 10: 62.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 10000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 10000000.0)
   0.5
   (/ 1.0 (* (/ (* (* x x) x) (* (* s s) s)) -0.16666666666666666))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 10000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((((x * x) * x) / ((s * s) * s)) * -0.16666666666666666f);
	}
	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) <= 10000000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((x * x) * x) / ((s * s) * s)) * (-0.16666666666666666e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(10000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) * x) / Float32(Float32(s * s) * s)) * Float32(-0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(10000000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((((x * x) * x) / ((s * s) * s)) * single(-0.16666666666666666));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1e7

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 48.6%

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

   if 1e7 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      3. lower--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\frac{-1}{6} \cdot \color{blue}{\frac{{x}^{3}}{{s}^{3}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.2%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 10000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot x}{\left(s \cdot s\right) \cdot s} \cdot -0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf((x / s), fmaf((0.5f / s), x, -1.0f), 2.0f);
	}
	return tmp;
}

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 82.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{0.16666666666666666}{\left(s \cdot s\right) \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.0000000036274937e-15)
   0.5
   (/ 1.0 (- 2.0 (* (* (* x x) x) (/ 0.16666666666666666 (* (* s s) s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.0000000036274937e-15f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (((x * x) * x) * (0.16666666666666666f / ((s * s) * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 1.0000000036274937e-15) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (((x * x) * x) * (0.16666666666666666e0 / ((s * s) * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.0000000036274937e-15))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(Float32(x * x) * x) * Float32(Float32(0.16666666666666666) / Float32(Float32(s * s) * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.0000000036274937e-15))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (((x * x) * x) * (single(0.16666666666666666) / ((s * s) * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1e-15

   1. Initial program 99.8%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 48.7%

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

   if 1e-15 < (neg.f32 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      3. lower--.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 - \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{2 - \color{blue}{\frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in s around inf

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

   8. Applied rewrites 78.1%

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

   9. Taylor expanded in s around 0

      \[\leadsto \frac{1}{2 - \frac{1}{6} \cdot \color{blue}{\frac{{x}^{3}}{{s}^{3}}}} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{0.16666666666666666}{\left(s \cdot s\right) \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10000.0f) {
		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) <= (-10000.0e0)) 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(-10000.0))
		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(-10000.0))
		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) < -1e4

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 59.0

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

   5. Applied rewrites 59.0%

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

Alternative 14: 34.9% 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 99.8%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 35.7%

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

Reproduce

herbie shell --seed 2024235 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))