Logistic function

Percentage Accurate: 99.8% → 99.5%

Time: 7.4s

Alternatives: 17

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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. pow2 N/A

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

   7. lift-exp.f32 N/A

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

   8. sinh-+-cosh-rev N/A

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

   9. flip-+ N/A

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

   10. sinh-cosh N/A

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

   11. sinh---cosh-rev N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-pow.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. pow2 N/A

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

   7. lift-exp.f32 N/A

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

   8. sinh-+-cosh-rev N/A

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

   9. flip-+ N/A

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

   10. sinh-cosh N/A

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

   11. sinh---cosh-rev N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-pow.f32 N/A

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

4. Applied rewrites 99.6%

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

Alternative 3: 92.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.4%

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

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 85.1%

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

   9. Applied rewrites 91.4%

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

   if 0.600000024 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 27.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

   9. Applied rewrites 99.6%

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

Alternative 4: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, \frac{x \cdot \mathsf{fma}\left(0.5, x, -s\right)}{s \cdot s}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5)
   (/ 1.0 (fma 1.0 (/ (* x (fma 0.5 x (- s))) (* s s)) 1.0))
   (/ 1.0 (- 2.0 (* (* x (/ (/ x s) s)) -0.5)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 1.0f / fmaf(1.0f, ((x * fmaf(0.5f, x, -s)) / (s * s)), 1.0f);
	} else {
		tmp = 1.0f / (2.0f - ((x * ((x / s) / s)) * -0.5f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(x * fma(Float32(0.5), x, Float32(-s))) / Float32(s * s)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x * Float32(Float32(x / s) / s)) * Float32(-0.5))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 28.1%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 6.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   10. Applied rewrites 99.6%

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

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. pow2 N/A

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

      7. lift-exp.f32 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. flip-+ N/A

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

      10. sinh-cosh N/A

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

      11. sinh---cosh-rev N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.0%

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

Alternative 5: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5)
   (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))
   (/ 1.0 (- 2.0 (* (* x (/ (/ x s) s)) -0.5)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	} else {
		tmp = 1.0f / (2.0f - ((x * ((x / s) / s)) * -0.5f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / s)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x * Float32(Float32(x / s) / s)) * Float32(-0.5))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 98.6%

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

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. pow2 N/A

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

      7. lift-exp.f32 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. flip-+ N/A

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

      10. sinh-cosh N/A

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

      11. sinh---cosh-rev N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 61.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 82.0%

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

Alternative 6: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5)
   (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / s)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 98.6%

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

   if 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.4%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 61.6

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

   5. Applied rewrites 61.6%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. pow2 N/A

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

   7. lift-exp.f32 N/A

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

   8. sinh-+-cosh-rev N/A

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

   9. flip-+ N/A

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

   10. sinh-cosh N/A

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

   11. sinh---cosh-rev N/A

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

   12. inv-pow N/A

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

   13. pow-pow N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. lower-pow.f32 N/A

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

4. Applied rewrites 99.6%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. sqrt-pow1 N/A

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

   4. metadata-eval N/A

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

   5. unpow-1 N/A

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

   6. lift-pow.f32 N/A

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

   7. pow-flip N/A

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

   8. lift-/.f32 N/A

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

   9. distribute-frac-neg N/A

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

   10. lower-pow.f32 N/A

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

   11. lift-exp.f32 N/A

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

   12. exp-1-e N/A

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

   13. lower-E.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower-neg.f32 99.6

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

8. Applied rewrites 99.6%

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

Alternative 8: 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 99.6%

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

Alternative 9: 92.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -5.0)
   (/ 1.0 (fma 1.0 (fma (- (* 0.5 (/ x s)) 1.0) (/ x s) 1.0) 1.0))
   (/
    1.0
    (+
     1.0
     (+
      (* (/ (- (/ (* (* -0.16666666666666666 (/ x s)) x) s) 1.0) s) x)
      1.0)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, fmaf(((0.5f * (x / s)) - 1.0f), (x / s), 1.0f), 1.0f);
	} else {
		tmp = 1.0f / (1.0f + (((((((-0.16666666666666666f * (x / s)) * x) / s) - 1.0f) / s) * x) + 1.0f));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), fma(Float32(Float32(Float32(0.5) * Float32(x / s)) - Float32(1.0)), Float32(x / s), Float32(1.0)), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(Float32(Float32(Float32(Float32(-0.16666666666666666) * Float32(x / s)) * x) / s) - Float32(1.0)) / s) * x) + Float32(1.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

   9. Applied rewrites 99.6%

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

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

   1. Initial program 99.4%

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

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 36.1%

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

   7. Applied rewrites 85.1%

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

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{1 + \left(\frac{\frac{\left(\frac{-1}{6} \cdot \frac{x}{s}\right) \cdot x}{s} - 1}{s} \cdot x + 1\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 90.0%

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

Alternative 10: 88.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 100000:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{x \cdot \left(\left(-s\right) + 0.5 \cdot x\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))
     (if (<= t_0 100000.0)
       (/ 1.0 (- 2.0 (/ x s)))
       (/ 1.0 (+ 1.0 (/ (* x (+ (- s) (* 0.5 x))) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	} else if (t_0 <= 100000.0f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + ((x * (-s + (0.5f * x))) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / s)), Float32(1.0)));
	elseif (t_0 <= Float32(100000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(x * Float32(Float32(-s) + Float32(Float32(0.5) * x))) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 99.6%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1e5

   1. Initial program 98.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 83.0

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

   5. Applied rewrites 83.0%

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

   if 1e5 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 86.8%

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

   10. Applied rewrites 86.8%

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

Alternative 11: 88.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 100000:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))
     (if (<= t_0 100000.0)
       (/ 1.0 (- 2.0 (/ x s)))
       (/ 1.0 (+ 1.0 (/ (* (* 0.5 x) x) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	} else if (t_0 <= 100000.0f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + (((0.5f * x) * x) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / s)), Float32(1.0)));
	elseif (t_0 <= Float32(100000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(Float32(0.5) * x) * x) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 98.6%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1e5

   1. Initial program 98.5%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 83.0

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

   5. Applied rewrites 83.0%

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

   if 1e5 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 86.8%

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

Alternative 12: 79.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, 1 - \frac{x}{s}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 500000000:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{\left(-x\right) \cdot s}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (/ 1.0 (fma 1.0 (- 1.0 (/ x s)) 1.0))
     (if (<= t_0 500000000.0)
       (/ 1.0 (- 2.0 (/ x s)))
       (/ 1.0 (+ 1.0 (/ (* (- x) s) (* s s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / fmaf(1.0f, (1.0f - (x / s)), 1.0f);
	} else if (t_0 <= 500000000.0f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f / (1.0f + ((-x * s) / (s * s)));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), Float32(Float32(1.0) - Float32(x / s)), Float32(1.0)));
	elseif (t_0 <= Float32(500000000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(Float32(-x) * s) / Float32(s * s))));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.1

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

   5. Applied rewrites 5.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 98.6%

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

   if -5 < (/.f32 (neg.f32 x) s) < 5e8

   1. Initial program 98.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 77.9

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

   5. Applied rewrites 77.9%

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

   if 5e8 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 62.7%

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

Alternative 13: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 27.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.6

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

   7. Applied rewrites 99.6%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

   9. Applied rewrites 99.6%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 36.5

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 85.1%

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

Alternative 14: 89.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 6.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   10. Applied rewrites 99.6%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 36.5

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 85.1%

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

Alternative 15: 86.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 27.9%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 6.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 99.6

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

   10. Applied rewrites 99.6%

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

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

   1. Initial program 99.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. pow2 N/A

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

      7. lift-exp.f32 N/A

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

      8. sinh-+-cosh-rev N/A

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

      9. flip-+ N/A

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

      10. sinh-cosh N/A

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

      11. sinh---cosh-rev N/A

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

      12. inv-pow N/A

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

      13. pow-pow N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-pow.f32 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in s around inf

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

   7. Applied rewrites 61.0%

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

   9. Applied rewrites 83.4%

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

Alternative 16: 49.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -5.0f) {
		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) <= (-5.0e0)) 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(-5.0))
		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(-5.0))
		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) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.4%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 61.6

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

   5. Applied rewrites 61.6%

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

Alternative 17: 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 99.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 33.4%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))