Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.2s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.4× 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 99.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. *-commutative N/A

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

   8. log-pow N/A

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

   9. inv-pow N/A

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

   10. lift-/.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. log-rec N/A

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

   14. lower-neg.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(e^{\frac{x \cdot 1.5}{s \cdot -2}}, e^{-0.25 \cdot \frac{x}{s}}, 1\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (fma (exp (/ (* x 1.5) (* s -2.0))) (exp (- (* 0.25 (/ x s)))) 1.0)))

C

float code(float x, float s) {
	return 1.0f / fmaf(expf(((x * 1.5f) / (s * -2.0f))), expf(-(0.25f * (x / s))), 1.0f);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / fma(exp(Float32(Float32(x * Float32(1.5)) / Float32(s * Float32(-2.0)))), exp(Float32(-Float32(Float32(0.25) * Float32(x / s)))), Float32(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-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. *-lft-identity N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f32 N/A

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

   6. exp-1-e N/A

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

   7. lower-E.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-E.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-pow.f32 N/A

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

   7. sqr-pow N/A

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

   8. pow-prod-down N/A

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

   9. lift-E.f32 N/A

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

   10. add-sqr-sqrt N/A

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

   11. associate-*r* N/A

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

   12. unpow-prod-down N/A

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

   13. lower-fma.f32 N/A

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.7%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(e^{\frac{1.5 \cdot x}{-2 \cdot s}}, e^{0.25 \cdot \frac{x}{-s}}, 1\right)}} \]

9. Final simplification 99.7%

   \[\leadsto \frac{1}{\mathsf{fma}\left(e^{\frac{x \cdot 1.5}{s \cdot -2}}, e^{-0.25 \cdot \frac{x}{s}}, 1\right)} \]
10. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(-x) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (-x / s)));
end

Derivation

1. Initial program 99.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. *-lft-identity N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f32 N/A

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

   6. exp-1-e N/A

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

   7. lower-E.f32 99.7

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

4. Applied rewrites 99.7%

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

Alternative 4: 99.8% 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: 67.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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(1.0) + fma(x, fma(Float32(Float32(1.0) / s), Float32(fma(Float32(x / s), Float32(-0.16666666666666666), Float32(0.5)) / Float32(s / x)), Float32(Float32(-1.0) / s)), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. 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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 86.0%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. clear-num N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f32 N/A

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

      10. associate-/l* N/A

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

      11. times-frac N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-/.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lift-fma.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

      18. lower-/.f32 90.0

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

   7. Applied rewrites 90.0%

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

Alternative 6: 67.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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(1.0) + fma(x, fma(Float32(fma(Float32(x / s), Float32(-0.16666666666666666), Float32(0.5)) / s), Float32(x / s), Float32(Float32(-1.0) / s)), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. 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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 86.0%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f32 N/A

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

      10. times-frac N/A

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

      11. lift-/.f32 N/A

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

      12. lower-fma.f32 N/A

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

      13. lower-/.f32 90.0

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

      14. lift-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f32 90.0

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

   7. Applied rewrites 90.0%

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

Alternative 7: 64.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.0000000195414814 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, x \cdot -0.16666666666666666, s \cdot \mathsf{fma}\left(0.5, x, -s\right)\right)}{s \cdot \left(s \cdot s\right)}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.0000000195414814e-24)
   0.5
   (/
    1.0
    (+
     1.0
     (fma
      x
      (/
       (fma x (* x -0.16666666666666666) (* s (fma 0.5 x (- s))))
       (* s (* s s)))
      1.0)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.0000000195414814e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + fmaf(x, (fmaf(x, (x * -0.16666666666666666f), (s * fmaf(0.5f, x, -s))) / (s * (s * s))), 1.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.0000000195414814e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(x, Float32(fma(x, Float32(x * Float32(-0.16666666666666666)), Float32(s * fma(Float32(0.5), x, Float32(-s)))) / Float32(s * Float32(s * s))), Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.00000002e-24

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 47.4%

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

   if 1.00000002e-24 < (neg.f32 x)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-fma.f32 N/A

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

      8. distribute-rgt-out N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f32 N/A

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

      16. cube-mult N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f32 90.9

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

   8. Applied rewrites 90.9%

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

Alternative 8: 66.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   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 50.6%

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

   if 50 < (/.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

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. lower-*.f32 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f32 N/A

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

      20. cube-mult N/A

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

      21. unpow2 N/A

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

      22. lower-*.f32 N/A

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

      23. unpow2 N/A

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

      24. lower-*.f32 94.0

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

   8. Applied rewrites 94.0%

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

Alternative 9: 63.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 50.6%

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

   if 50 < (/.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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 76.0

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

   8. Applied rewrites 76.0%

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

   9. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. associate-*l/ N/A

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. associate-*r/ N/A

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

       10. metadata-eval N/A

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

       11. lower-/.f32 N/A

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

       12. unpow2 N/A

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

       13. lower-*.f32 87.9

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

   11. Applied rewrites 87.9%

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

4. Final simplification 63.6%

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

Alternative 10: 60.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.2%

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

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

   1. Initial program 99.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 82.3

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

   8. Applied rewrites 82.3%

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

Alternative 11: 48.9% 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 99.9%

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

   3. Taylor expanded in x around 0

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

   5. 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 65.0

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

   5. Applied rewrites 65.0%

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

Alternative 12: 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 35.2%

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))