Logistic function

Percentage Accurate: 99.8% → 99.7%

Time: 9.0s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{s} \cdot -0.5\\ \frac{1}{{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{t\_0} \cdot {\left(e^{1.6666666666666667}\right)}^{t\_0} + 1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (* (/ x s) -0.5)))
   (/
    1.0
    (+ (* (pow (cbrt (E)) t_0) (pow (exp 1.6666666666666667) t_0)) 1.0))))

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%

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

6. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(\mathsf{E}\left(\right) \cdot e^{0.6666666666666666}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}}} \]
7. Step-by-step derivation

   1. rem-exp-log N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\log \left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}\right)}}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   2. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\log \left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}\right)}}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \color{blue}{\left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   4. lift-E.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(\color{blue}{\mathsf{E}\left(\right)} \cdot e^{\frac{2}{3}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   5. e-exp-1 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(\color{blue}{e^{1}} \cdot e^{\frac{2}{3}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   6. lift-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(e^{1} \cdot \color{blue}{e^{\frac{2}{3}}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   7. prod-exp N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \color{blue}{\left(e^{1 + \frac{2}{3}}\right)}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   8. rem-log-exp N/A

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

   9. metadata-eval 99.8

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{1.6666666666666667}}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{1.6666666666666667}\right)}}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{1}{{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} \cdot {\left(e^{1.6666666666666667}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} + 1} \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left(e^{-0.16666666666666666}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{1.6666666666666667}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   (*
    (pow (exp -0.16666666666666666) (/ x s))
    (pow (exp 1.6666666666666667) (* (/ x s) -0.5)))
   1.0)))

C

float code(float x, float s) {
	return 1.0f / ((powf(expf(-0.16666666666666666f), (x / s)) * powf(expf(1.6666666666666667f), ((x / s) * -0.5f))) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (((exp((-0.16666666666666666e0)) ** (x / s)) * (exp(1.6666666666666667e0) ** ((x / s) * (-0.5e0)))) + 1.0e0)
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32((exp(Float32(-0.16666666666666666)) ^ Float32(x / s)) * (exp(Float32(1.6666666666666667)) ^ Float32(Float32(x / s) * Float32(-0.5)))) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (((exp(single(-0.16666666666666666)) ^ (x / s)) * (exp(single(1.6666666666666667)) ^ ((x / s) * single(-0.5)))) + single(1.0));
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%

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

6. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(\mathsf{E}\left(\right) \cdot e^{0.6666666666666666}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}}} \]
7. Step-by-step derivation

   1. rem-exp-log N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\log \left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}\right)}}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   2. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{\log \left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}\right)}}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \color{blue}{\left(\mathsf{E}\left(\right) \cdot e^{\frac{2}{3}}\right)}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   4. lift-E.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(\color{blue}{\mathsf{E}\left(\right)} \cdot e^{\frac{2}{3}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   5. e-exp-1 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(\color{blue}{e^{1}} \cdot e^{\frac{2}{3}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   6. lift-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \left(e^{1} \cdot \color{blue}{e^{\frac{2}{3}}}\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   7. prod-exp N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\log \color{blue}{\left(e^{1 + \frac{2}{3}}\right)}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}} \]

   8. rem-log-exp N/A

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

   9. metadata-eval 99.8

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{1.6666666666666667}}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}} \]

8. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + {\color{blue}{\left(e^{1.6666666666666667}\right)}}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)}} \]
9. Step-by-step derivation

   1. lift-pow.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\color{blue}{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)}}} \]

   3. pow-unpow N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot \color{blue}{{\left({\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}}} \]

   4. lower-pow.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot \color{blue}{{\left({\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}}} \]

   5. pow-to-exp N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\color{blue}{\left(e^{\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \frac{-1}{2}}\right)}}^{\left(\frac{x}{s}\right)}} \]

   6. lower-exp.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\color{blue}{\left(e^{\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \frac{-1}{2}}\right)}}^{\left(\frac{x}{s}\right)}} \]

   7. lift-cbrt.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)} \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   8. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\log \color{blue}{\left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)} \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   9. log-pow N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{\left(\frac{1}{3} \cdot \log \mathsf{E}\left(\right)\right)} \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   10. lift-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\left(\frac{1}{3} \cdot \log \color{blue}{\mathsf{E}\left(\right)}\right) \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   11. log-E N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\left(\frac{1}{3} \cdot \color{blue}{1}\right) \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{5}{3}}\right)}^{\left(\frac{-1}{2} \cdot \frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{\frac{1}{3}} \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   13. metadata-eval 99.8

       \[\leadsto \frac{1}{1 + {\left(e^{1.6666666666666667}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{-0.16666666666666666}}\right)}^{\left(\frac{x}{s}\right)}} \]

10. Applied rewrites 99.8%

    \[\leadsto \frac{1}{1 + {\left(e^{1.6666666666666667}\right)}^{\left(-0.5 \cdot \frac{x}{s}\right)} \cdot \color{blue}{{\left(e^{-0.16666666666666666}\right)}^{\left(\frac{x}{s}\right)}}} \]

11. Final simplification 99.8%

    \[\leadsto \frac{1}{{\left(e^{-0.16666666666666666}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{1.6666666666666667}\right)}^{\left(\frac{x}{s} \cdot -0.5\right)} + 1} \]
12. Add Preprocessing

Alternative 3: 62.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.5%

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

   10. Applied rewrites 77.6%

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

4. Final simplification 61.1%

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

Alternative 4: 48.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.4000000059604645:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.4000000059604645)
   0.5
   (/ 1.0 (+ (- 1.0 (/ x s)) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.4000000059604645f) {
		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 (exp((-x / s)) <= 0.4000000059604645e0) 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 (exp(Float32(Float32(-x) / s)) <= Float32(0.4000000059604645))
		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 (exp((-x / s)) <= single(0.4000000059604645))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 0.400000006

   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 0.400000006 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. 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 63.0

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

   5. Applied rewrites 63.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.4000000059604645:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(1 - \frac{x}{s}\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{-x}{s}} \leq 0.4000000059604645:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ (- x) s)) 0.4000000059604645) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (expf((-x / s)) <= 0.4000000059604645f) {
		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 (exp((-x / s)) <= 0.4000000059604645e0) 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 (exp(Float32(Float32(-x) / s)) <= Float32(0.4000000059604645))
		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 (exp((-x / s)) <= single(0.4000000059604645))
		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 (exp.f32 (/.f32 (neg.f32 x) s)) < 0.400000006

   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 0.400000006 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 62.9

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

   5. Applied rewrites 62.9%

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

Alternative 6: 47.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;e^{t\_0} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= (exp t_0) 2.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (expf(t_0) <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (exp(t_0) <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (exp(t_0) <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (exp(t_0) <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 39.0%

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

Alternative 7: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

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. Final simplification 99.8%

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

Alternative 8: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 9: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (- (/ 0.5 (* s s)) (/ (- (/ 1.0 s) (/ 2.0 x)) x)) x) x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f / (s * s)) - (((1.0f / s) - (2.0f / x)) / x)) * x) * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.10000000149011612e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / ((((0.5e0 / (s * s)) - (((1.0e0 / s) - (2.0e0 / x)) / x)) * x) * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		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(Float32(0.5) / Float32(s * s)) - Float32(Float32(Float32(Float32(1.0) / s) - Float32(Float32(2.0) / x)) / x)) * x) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		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)) - (((single(1.0) / s) - (single(2.0) / x)) / x)) * x) * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + \frac{x}{s}}}} \]
   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 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 6.6%

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

   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 79.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{0.5}{s \cdot s} - \frac{\frac{1}{s} - \frac{2}{x}}{x}\right) \cdot x\right) \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{\frac{1}{s}}{s} \cdot x\right) \cdot x\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (* (/ (/ 1.0 s) s) x) x) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / (((((1.0f / s) / s) * x) * x) * 0.5f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.10000000149011612e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / (((((1.0e0 / s) / s) * x) * x) * 0.5e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		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(Float32(Float32(1.0) / s) / s) * x) * x) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / (((((single(1.0) / s) / s) * x) * x) * single(0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + \frac{x}{s}}}} \]
   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 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.5%

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

   10. Applied rewrites 79.6%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(\frac{\frac{1}{s}}{s} \cdot x\right) \cdot x\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   (/ 1.0 (+ (/ 1.0 (+ (/ x s) 1.0)) 1.0))
   (/ 1.0 (* (* (/ x (* s s)) x) 0.5))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 1.0f / ((1.0f / ((x / s) + 1.0f)) + 1.0f);
	} else {
		tmp = 1.0f / (((x / (s * s)) * x) * 0.5f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.10000000149011612e0) then
        tmp = 1.0e0 / ((1.0e0 / ((x / s) + 1.0e0)) + 1.0e0)
    else
        tmp = 1.0e0 / (((x / (s * s)) * x) * 0.5e0)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		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(x / Float32(s * s)) * x) * Float32(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		tmp = single(1.0) / ((single(1.0) / ((x / s) + single(1.0))) + single(1.0));
	else
		tmp = single(1.0) / (((x / (s * s)) * x) * single(0.5));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   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. 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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \frac{1}{\color{blue}{1 + \frac{x}{s}}}} \]
   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 95.5

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

   7. Applied rewrites 95.5%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 6.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.5%

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

   10. Applied rewrites 77.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{s} + 1} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{x}{s \cdot s} \cdot x\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 34.6% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 36.1%

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

Reproduce

herbie shell --seed 2024295 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))