Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(2.0f), (-0.5f * (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(2.0e0) ** ((-0.5e0) * (x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-exp.f32 N/A

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

   2. *-lft-identity N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. exp-1-e N/A

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

   6. lower-E.f32 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-pow.f32 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lift-E.f32 N/A

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

   5. lift-E.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. e-exp-1 N/A

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

   8. e-exp-1 N/A

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

   9. prod-exp N/A

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

   10. metadata-eval N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. associate-/l/ N/A

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

   14. lift-neg.f32 N/A

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

   15. neg-mul-1 N/A

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

   16. times-frac N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f32 N/A

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

   20. lower-*.f32 N/A

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

   21. metadata-eval 99.9

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

6. Applied rewrites 99.9%

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

Alternative 2: 58.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 5.000000015855384 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x - \left(\frac{0.5}{s} \cdot x\right) \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 5.000000015855384e-29)
   (/ 1.0 (+ 1.0 (fma (/ x s) (fma (/ 0.5 s) x -1.0) 1.0)))
   (/ 1.0 (- 2.0 (/ (- x (* (* (/ 0.5 s) x) x)) s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 5.000000015855384e-29f) {
		tmp = 1.0f / (1.0f + fmaf((x / s), fmaf((0.5f / s), x, -1.0f), 1.0f));
	} else {
		tmp = 1.0f / (2.0f - ((x - (((0.5f / s) * x) * x)) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(5.000000015855384e-29))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(x / s), fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(1.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x - Float32(Float32(Float32(Float32(0.5) / s) * x) * x)) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 5.00000002e-29

   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. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 28.1%

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

   if 5.00000002e-29 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 82.4%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 5.000000015855384 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x - \left(\frac{0.5}{s} \cdot x\right) \cdot x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 4: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ t_1 := \frac{0.5}{s \cdot s}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(t\_1, x, \frac{-1}{s}\right), x, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 100:\\ \;\;\;\;\frac{1}{2 - \frac{x - \left(\frac{0.5}{s} \cdot x\right) \cdot x}{s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(t\_1 - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)) (t_1 (/ 0.5 (* s s))))
   (if (<= t_0 -50.0)
     (/ 1.0 (+ 1.0 (fma (fma t_1 x (/ -1.0 s)) x 1.0)))
     (if (<= t_0 100.0)
       (/ 1.0 (- 2.0 (/ (- x (* (* (/ 0.5 s) x) x)) s)))
       (/ 1.0 (* (* (- t_1 (/ (- (/ 1.0 s) (/ 2.0 x)) x)) x) x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float t_1 = 0.5f / (s * s);
	float tmp;
	if (t_0 <= -50.0f) {
		tmp = 1.0f / (1.0f + fmaf(fmaf(t_1, x, (-1.0f / s)), x, 1.0f));
	} else if (t_0 <= 100.0f) {
		tmp = 1.0f / (2.0f - ((x - (((0.5f / s) * x) * x)) / s));
	} else {
		tmp = 1.0f / (((t_1 - (((1.0f / s) - (2.0f / x)) / x)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	t_1 = Float32(Float32(0.5) / Float32(s * s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(fma(t_1, x, Float32(Float32(-1.0) / s)), x, Float32(1.0))));
	elseif (t_0 <= Float32(100.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x - Float32(Float32(Float32(Float32(0.5) / s) * x) * x)) / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(t_1 - Float32(Float32(Float32(Float32(1.0) / s) - Float32(Float32(2.0) / x)) / x)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f32 N/A

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

      5. exp-1-e N/A

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

      6. lower-E.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s) < 100

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 87.7%

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 94.0%

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

   if 100 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 85.1%

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

Alternative 5: 59.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -50:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}\\ \mathbf{elif}\;t\_0 \leq 500000:\\ \;\;\;\;\frac{1}{2 - \frac{1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -50.0)
     (/ 1.0 (+ 1.0 (fma (/ x s) (fma (/ 0.5 s) x -1.0) 1.0)))
     (if (<= t_0 500000.0)
       (/ 1.0 (- 2.0 (/ 1.0 (/ s x))))
       (/ 1.0 (/ (* x (- (* 0.5 x) s)) (* s s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -50.0f) {
		tmp = 1.0f / (1.0f + fmaf((x / s), fmaf((0.5f / s), x, -1.0f), 1.0f));
	} else if (t_0 <= 500000.0f) {
		tmp = 1.0f / (2.0f - (1.0f / (s / x)));
	} else {
		tmp = 1.0f / ((x * ((0.5f * 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(-50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(Float32(x / s), fma(Float32(Float32(0.5) / s), x, Float32(-1.0)), Float32(1.0))));
	elseif (t_0 <= Float32(500000.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(1.0) / Float32(s / x))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(Float32(0.5) * x) - s)) / Float32(s * s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   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. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 28.1%

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

   if -50 < (/.f32 (neg.f32 x) s) < 5e5

   1. Initial program 99.5%

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

      1. lift-exp.f32 N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f32 N/A

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

      5. exp-1-e N/A

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

      6. lower-E.f32 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. log-E N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f32 N/A

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

      6. lower-/.f32 87.2

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

   7. Applied rewrites 87.2%

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

   9. Applied rewrites 87.2%

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

   if 5e5 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 45.4%

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

   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%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 1\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 500000:\\ \;\;\;\;\frac{1}{2 - \frac{1}{\frac{s}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5}{s \cdot s}, x, \frac{-1}{s}\right), x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x - \left(\frac{0.5}{s} \cdot x\right) \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -50.0)
   (/ 1.0 (+ 1.0 (fma (fma (/ 0.5 (* s s)) x (/ -1.0 s)) x 1.0)))
   (/ 1.0 (- 2.0 (/ (- x (* (* (/ 0.5 s) x) x)) s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -50.0f) {
		tmp = 1.0f / (1.0f + fmaf(fmaf((0.5f / (s * s)), x, (-1.0f / s)), x, 1.0f));
	} else {
		tmp = 1.0f / (2.0f - ((x - (((0.5f / s) * x) * x)) / s));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-50.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + fma(fma(Float32(Float32(0.5) / Float32(s * s)), x, Float32(Float32(-1.0) / s)), x, Float32(1.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(x - Float32(Float32(Float32(Float32(0.5) / s) * x) * x)) / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. lift-exp.f32 N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f32 N/A

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

      5. exp-1-e N/A

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

      6. lower-E.f32 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

   7. Applied rewrites 28.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 46.0%

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 82.4%

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

Alternative 7: 60.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 500000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 500000.0) 0.5 (/ 1.0 (/ (* x (- (* 0.5 x) s)) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 500000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((x * ((0.5f * x) - s)) / (s * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 500000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((x * ((0.5e0 * x) - s)) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(500000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(Float32(0.5) * x) - s)) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(500000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((x * ((single(0.5) * x) - s)) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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 54.4%

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

   if 5e5 < (/.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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. associate-*r/ N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-fma.f32 N/A

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

   5. Applied rewrites 6.3%

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

   7. Applied rewrites 45.4%

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

   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%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 500000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{1}{s} \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -50.0) 0.5 (/ 1.0 (- 2.0 (* (/ 1.0 s) x)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -50.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - ((1.0f / s) * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-50.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - ((1.0e0 / s) * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-50.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(Float32(Float32(1.0) / s) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-50.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - ((single(1.0) / s) * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   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 -50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

      1. lift-exp.f32 N/A

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

      2. *-lft-identity N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f32 N/A

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

      5. exp-1-e N/A

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

      6. lower-E.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. log-E N/A

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

      4. *-rgt-identity N/A

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

      5. lower--.f32 N/A

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

      6. lower-/.f32 67.1

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

   7. Applied rewrites 67.1%

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

   9. Applied rewrites 67.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{1}{s} \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -50.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -50.0f) {
		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) <= (-50.0e0)) 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(-50.0))
		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(-50.0))
		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) < -50

   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 -50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 67.1

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

   5. Applied rewrites 67.1%

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

Alternative 10: 35.5% 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 39.6%

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

Reproduce

herbie shell --seed 2024322 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))