Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.9s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 89.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{s \cdot s}}\\ \mathbf{elif}\;t\_0 \leq 0.6000000238418579:\\ \;\;\;\;0.25 \cdot \frac{x}{s} + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5 \cdot \frac{x}{s} - 1}{s}, x, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 4.0000000467443897e-7)
     (/ 1.0 (/ (- (* (* x x) 0.5) (* s x)) (* s s)))
     (if (<= t_0 0.6000000238418579)
       (+ (* 0.25 (/ x s)) 0.5)
       (/ 1.0 (fma (fma (/ (- (* 0.5 (/ x s)) 1.0) s) x 1.0) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 4.0000000467443897e-7f) {
		tmp = 1.0f / ((((x * x) * 0.5f) - (s * x)) / (s * s));
	} else if (t_0 <= 0.6000000238418579f) {
		tmp = (0.25f * (x / s)) + 0.5f;
	} else {
		tmp = 1.0f / fmaf(fmaf((((0.5f * (x / s)) - 1.0f) / s), x, 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) * Float32(0.5)) - Float32(s * x)) / Float32(s * s)));
	elseif (t_0 <= Float32(0.6000000238418579))
		tmp = Float32(Float32(Float32(0.25) * Float32(x / s)) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / fma(fma(Float32(Float32(Float32(Float32(0.5) * Float32(x / s)) - Float32(1.0)) / s), x, Float32(1.0)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. lower--.f32 N/A

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

      13. lower-/.f32 N/A

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

   5. Applied rewrites 37.9%

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

   7. Applied rewrites 36.9%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.3%

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

   if 4.00000005e-7 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.600000024

   1. Initial program 99.6%

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

      1. lift-+.f32 N/A

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

      2. lift-exp.f32 N/A

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

      3. lift-/.f32 N/A

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

      4. lift-neg.f32 N/A

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

      5. distribute-frac-neg N/A

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

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

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

      7. cosh-neg N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f32 N/A

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

      10. lift-/.f32 N/A

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

      11. associate-+r- N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f32 N/A

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

      15. lift-/.f32 N/A

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

      16. lift-neg.f32 N/A

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

      17. distribute-frac-neg N/A

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

      18. cosh-neg N/A

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

      19. lower-cosh.f32 N/A

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

      20. lower-/.f32 N/A

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

      21. lower-sinh.f32 N/A

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

      22. lower-/.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{x - -1 \cdot x}{s} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x - -1 \cdot x}{s} \cdot \frac{1}{8}} + \frac{1}{2} \]

      3. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - -1 \cdot x}{s}, \frac{1}{8}, \frac{1}{2}\right)} \]

      4. lower-/.f32 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x - -1 \cdot x}{s}}, \frac{1}{8}, \frac{1}{2}\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x + \left(\mathsf{neg}\left(-1\right)\right) \cdot x}}{s}, \frac{1}{8}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{x + \color{blue}{1} \cdot x}{s}, \frac{1}{8}, \frac{1}{2}\right) \]

      7. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(1 + 1\right) \cdot x}}{s}, \frac{1}{8}, \frac{1}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2} \cdot x}{s}, \frac{1}{8}, \frac{1}{2}\right) \]

      9. lower-*.f32 89.0

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2 \cdot x}}{s}, 0.125, 0.5\right) \]

   7. Applied rewrites 87.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2 \cdot x}{s}, 0.125, 0.5\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 99.0%

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

   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. 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 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)\right)}} \]
   6. 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. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-/l* N/A

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

      7. div-sub N/A

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

      8. lower-/.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-/.f32 28.1

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

   7. Applied rewrites 27.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.4

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

   9. Applied rewrites 99.4%

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

Alternative 3: 87.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \sqrt{\frac{1}{1 - \frac{-2 \cdot \left(\frac{x \cdot x}{s} + x\right)}{s}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 4.0000000467443897e-7)
   (/ 1.0 (/ (- (* (* x x) 0.5) (* s x)) (* s s)))
   (/ 1.0 (+ 1.0 (sqrt (/ 1.0 (- 1.0 (/ (* -2.0 (+ (/ (* x x) s) x)) s))))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 4.0000000467443897e-7f) {
		tmp = 1.0f / ((((x * x) * 0.5f) - (s * x)) / (s * s));
	} else {
		tmp = 1.0f / (1.0f + sqrtf((1.0f / (1.0f - ((-2.0f * (((x * x) / s) + 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 ((1.0e0 / (1.0e0 + exp((-x / s)))) <= 4.0000000467443897e-7) then
        tmp = 1.0e0 / ((((x * x) * 0.5e0) - (s * x)) / (s * s))
    else
        tmp = 1.0e0 / (1.0e0 + sqrt((1.0e0 / (1.0e0 - (((-2.0e0) * (((x * x) / s) + x)) / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) * Float32(0.5)) - Float32(s * x)) / Float32(s * s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + sqrt(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(-2.0) * Float32(Float32(Float32(x * x) / s) + x)) / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) / (single(1.0) + exp((-x / s)))) <= single(4.0000000467443897e-7))
		tmp = single(1.0) / ((((x * x) * single(0.5)) - (s * x)) / (s * s));
	else
		tmp = single(1.0) / (single(1.0) + sqrt((single(1.0) / (single(1.0) - ((single(-2.0) * (((x * x) / s) + x)) / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 4.00000005e-7

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. lower--.f32 N/A

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

      13. lower-/.f32 N/A

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

   5. Applied rewrites 36.9%

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

   7. Applied rewrites 36.9%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 84.3%

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

   if 4.00000005e-7 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 99.8%

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

      1. lift-pow.f32 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. pow-prod-down N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f32 N/A

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

      7. lift-exp.f32 N/A

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

      8. lift-exp.f32 N/A

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

      9. exp-lft-sqr-rev N/A

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

      10. *-commutative N/A

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

      11. lift-/.f32 N/A

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

      12. associate-/l* N/A

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

      13. lift-*.f32 N/A

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

      14. lift-/.f32 N/A

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

      15. lower-exp.f32 99.8

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

      16. lift-/.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. associate-/l* N/A

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

      19. lift-/.f32 N/A

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

      20. lower-*.f32 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in s around -inf

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 95.5

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

   9. Applied rewrites 95.5%

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

Alternative 4: 61.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 50000.0)
   0.5
   (/ 1.0 (/ (- (* (* x x) 0.5) (* s x)) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((((x * x) * 0.5f) - (s * x)) / (s * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((((x * x) * 0.5e0) - (s * x)) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(x * x) * Float32(0.5)) - Float32(s * x)) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((((x * x) * single(0.5)) - (s * x)) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   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 50.5%

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

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. lower--.f32 N/A

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

      13. lower-/.f32 N/A

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 91.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left(x \cdot x\right) \cdot 0.5 - s \cdot x}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.03999999910593033:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.03999999910593033)
   0.5
   (/ 1.0 (* (* x (/ (/ x s) s)) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.03999999910593033f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * ((x / s) / s)) * 0.5f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.03999999910593033e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * ((x / s) / s)) * 0.5e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.03999999910593033))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(x / s) / s)) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.03999999910593033))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x * ((x / s) / s)) * single(0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0399999991

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.2%

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

   if 0.0399999991 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. lower--.f32 N/A

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

      13. lower-/.f32 N/A

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

   5. Applied rewrites 36.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 80.4%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.03999999910593033:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} \cdot \frac{0.5}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 50000.0) 0.5 (/ 1.0 (* (/ (* x x) s) (/ 0.5 s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((x * x) / s) * (0.5f / 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) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((x * x) / s) * (0.5e0 / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / s) * Float32(Float32(0.5) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((x * x) / s) * (single(0.5) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   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 50.5%

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

   if 5e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. associate-+l- N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

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

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

      12. lower--.f32 N/A

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

      13. lower-/.f32 N/A

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

   5. Applied rewrites 39.4%

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

   7. Applied rewrites 39.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot x}{s} \cdot \frac{0.5}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 63.6

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

   5. Applied rewrites 63.6%

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

4. Final simplification 49.5%

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

Alternative 8: 49.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. 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 63.6

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

   5. Applied rewrites 63.6%

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

4. Final simplification 49.5%

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

Alternative 9: 34.9% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 36.7%

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

6. Final simplification 36.7%

   \[\leadsto 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))