Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

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, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / ((1.0f / powf(expf(2.0f), ((x * 0.5f) / s))) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((1.0e0 / (exp(2.0e0) ** ((x * 0.5e0) / s))) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. pow-exp N/A

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

   8. e-exp-1 N/A

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

   9. lift-E.f32 N/A

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

   10. rem-square-sqrt N/A

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

   11. sqrt-unprod N/A

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

   12. lift-E.f32 N/A

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

   13. lift-E.f32 N/A

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

   14. sqrt-pow2 N/A

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

   15. lower-pow.f32 N/A

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

   16. e-exp-1 N/A

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

   17. e-exp-1 N/A

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

   18. prod-exp N/A

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

   19. metadata-eval N/A

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

   20. lower-exp.f32 N/A

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

   21. lower-/.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. lift-/.f32 N/A

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

   6. associate-/l* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (powf(expf(2.0f), (-0.5f * (x / s))) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((exp(2.0e0) ** ((-0.5e0) * (x / s))) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. clear-num N/A

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

   6. lift-/.f32 N/A

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

   7. pow-exp N/A

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

   8. e-exp-1 N/A

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

   9. lift-E.f32 N/A

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

   10. rem-square-sqrt N/A

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

   11. sqrt-unprod N/A

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

   12. lift-E.f32 N/A

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

   13. lift-E.f32 N/A

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

   14. sqrt-pow2 N/A

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

   15. lower-pow.f32 N/A

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

   16. e-exp-1 N/A

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

   17. e-exp-1 N/A

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

   18. prod-exp N/A

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

   19. metadata-eval N/A

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

   20. lower-exp.f32 N/A

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

   21. lower-/.f32 99.7

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

6. Applied rewrites 99.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f32 99.7

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

   4. lift-/.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. pow-flip N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f32 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 199999995904:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.5 - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 199999995904.0)
   (/ 1.0 (+ (/ 1.0 (- 1.0 (/ (- (* (* (/ x s) x) -0.5) x) s))) 1.0))
   (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 199999995904.0f) {
		tmp = 1.0f / ((1.0f / (1.0f - (((((x / s) * x) * -0.5f) - x) / s))) + 1.0f);
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 199999995904.0e0) then
        tmp = 1.0e0 / ((1.0e0 / (1.0e0 - (((((x / s) * x) * (-0.5e0)) - x) / s))) + 1.0e0)
    else
        tmp = 1.0e0 / (((0.5e0 / (s * s)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(199999995904.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(x / s) * x) * Float32(-0.5)) - x) / s))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(199999995904.0))
		tmp = single(1.0) / ((single(1.0) / (single(1.0) - (((((x / s) * x) * single(-0.5)) - x) / s))) + single(1.0));
	else
		tmp = single(1.0) / (((single(0.5) / (s * s)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 199999996000

   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. unpow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f32 N/A

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

      14. lower-/.f32 95.0

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

   7. Applied rewrites 95.0%

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

   if 199999996000 < (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 s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 84.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 199999995904:\\ \;\;\;\;\frac{1}{\frac{1}{1 - \frac{\left(\frac{x}{s} \cdot x\right) \cdot -0.5 - x}{s}} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 199999995904:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 199999995904.0)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 199999995904.0f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 199999995904.0e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / (((0.5e0 / (s * s)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(199999995904.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) / Float32(Float32(x / s) + Float32(1.0))) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(199999995904.0))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / (((single(0.5) / (s * s)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 199999996000

   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. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. lower-/.f32 93.1

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

   7. Applied rewrites 93.1%

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

   if 199999996000 < (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 s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 84.3%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 199999995904:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 199999995904:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 199999995904.0)
   0.5
   (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 199999995904.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp((-x / s)) <= 199999995904.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((0.5e0 / (s * s)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(Float32(-x) / s)) <= Float32(199999995904.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp((-x / s)) <= single(199999995904.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((single(0.5) / (s * s)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 199999996000

   1. Initial program 99.7%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.6%

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

   if 199999996000 < (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 s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.4%

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

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 84.3%

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

Alternative 6: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.7

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 8: 49.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) 0.5 (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((1.0f - (x / s)) + 1.0f);
	}
	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) <= (-10.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((1.0e0 - (x / s)) + 1.0e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-10.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(1.0) - Float32(x / s)) + Float32(1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-10.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(1.0) - (x / s)) + single(1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 60.0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 47.6%

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

Alternative 9: 49.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -10.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -10.0f) {
		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) <= (-10.0e0)) 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(-10.0))
		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(-10.0))
		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) < -10

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 28.1%

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

   if -10 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 60.0

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

   5. Applied rewrites 60.0%

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

Alternative 10: 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 99.7%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 35.8%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))