Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.2s

Alternatives: 13

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{0.25}{s}}\right)}^{2} \cdot {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+ 1.0 (* (pow (exp (* (- x) (/ 0.25 s))) 2.0) (pow (exp -0.5) (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (powf(expf((-x * (0.25f / s))), 2.0f) * powf(expf(-0.5f), (x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + ((exp((-x * (0.25e0 / s))) ** 2.0e0) * (exp((-0.5e0)) ** (x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((exp(Float32(Float32(-x) * Float32(Float32(0.25) / s))) ^ Float32(2.0)) * (exp(Float32(-0.5)) ^ Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((exp((-x * (single(0.25) / s))) ^ single(2.0)) * (exp(single(-0.5)) ^ (x / s))));
end

Derivation

1. Initial program 99.7%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-sqrt.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. rem-square-sqrt N/A

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

   2. pow1/2 N/A

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

   3. pow1/2 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. pow1/2 N/A

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

   8. pow-pow N/A

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

   9. lift-exp.f32 N/A

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

   10. pow-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{-x}{s} \cdot \color{blue}{0.25}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{-x}{s} \cdot 0.25}\right)}^{2}} \cdot \sqrt{e^{\frac{-x}{s}}}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{-x}{s} \cdot \frac{1}{4}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{-x}{s}} \cdot \frac{1}{4}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{\left(-x\right) \cdot \frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   4. associate-/l* N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   6. lower-/.f32 99.7

      \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \color{blue}{\frac{0.25}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

8. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{0.25}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

9. Taylor expanded in x around inf

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

    1. exp-sqrt-rev N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot \color{blue}{e^{\frac{-1 \cdot \frac{x}{s}}{2}}}} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot e^{\frac{\color{blue}{\frac{x}{s} \cdot -1}}{2}}} \]

    3. associate-/l* N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot e^{\color{blue}{\frac{x}{s} \cdot \frac{-1}{2}}}} \]

    4. metadata-eval N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot e^{\frac{x}{s} \cdot \color{blue}{\frac{-1}{2}}}} \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot e^{\color{blue}{\frac{-1}{2} \cdot \frac{x}{s}}}} \]

    6. exp-prod N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot \color{blue}{{\left(e^{\frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}}} \]

    7. lower-pow.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot \color{blue}{{\left(e^{\frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}}} \]

    8. lower-exp.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}\right)}^{2} \cdot {\color{blue}{\left(e^{\frac{-1}{2}}\right)}}^{\left(\frac{x}{s}\right)}} \]

    9. lower-/.f32 99.8

       \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{0.25}{s}}\right)}^{2} \cdot {\left(e^{-0.5}\right)}^{\color{blue}{\left(\frac{x}{s}\right)}}} \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{0.25}{s}}\right)}^{2} \cdot \color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \frac{0.25}{s}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+ 1.0 (* (pow (exp (* (- x) (/ 0.25 s))) 2.0) (sqrt (exp (/ (- x) s)))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (powf(expf((-x * (0.25f / s))), 2.0f) * sqrtf(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 * (0.25e0 / s))) ** 2.0e0) * sqrt(exp((-x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((exp(Float32(Float32(-x) * Float32(Float32(0.25) / s))) ^ Float32(2.0)) * sqrt(exp(Float32(Float32(-x) / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((exp((-x * (single(0.25) / s))) ^ single(2.0)) * sqrt(exp((-x / s)))));
end

Derivation

1. Initial program 99.7%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. sqrt-prod N/A

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

   6. lower-*.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. lower-sqrt.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. rem-square-sqrt N/A

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

   2. pow1/2 N/A

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

   3. pow1/2 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f32 N/A

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

   6. lift-sqrt.f32 N/A

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

   7. pow1/2 N/A

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

   8. pow-pow N/A

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

   9. lift-exp.f32 N/A

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

   10. pow-exp N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. metadata-eval 99.7

       \[\leadsto \frac{1}{1 + {\left(e^{\frac{-x}{s} \cdot \color{blue}{0.25}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{-x}{s} \cdot 0.25}\right)}^{2}} \cdot \sqrt{e^{\frac{-x}{s}}}} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{-x}{s} \cdot \frac{1}{4}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{-x}{s}} \cdot \frac{1}{4}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\frac{\left(-x\right) \cdot \frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   4. associate-/l* N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{\frac{1}{4}}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   6. lower-/.f32 99.7

      \[\leadsto \frac{1}{1 + {\left(e^{\left(-x\right) \cdot \color{blue}{\frac{0.25}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

8. Applied rewrites 99.7%

   \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{\left(-x\right) \cdot \frac{0.25}{s}}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]
9. Add Preprocessing

Alternative 3: 75.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 98.1

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

   7. Applied rewrites 97.1%

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

   if 0.5 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 76.5%

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

Alternative 4: 75.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.2

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

   5. Applied rewrites 5.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. lower-fma.f32 98.1

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

   7. Applied rewrites 97.2%

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

   if 0.5 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 76.4%

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

Alternative 5: 48.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 0.5

   1. Initial program 99.9%

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

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.2

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

   5. Applied rewrites 27.9%

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

   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 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.2%

       \[\leadsto \frac{1}{\frac{2}{x} \cdot x} \]

   if 0.5 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   7. Applied rewrites 63.8%

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

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 63.8%

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

4. Final simplification 50.0%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

Alternative 7: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\frac{0.25}{s} \cdot x + 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{0.5}{s \cdot s} \cdot x\right) \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -200.0)
     (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))
     (if (<= t_0 0.05000000074505806)
       (+ (* (/ 0.25 s) x) 0.5)
       (/ 1.0 (* (* (/ 0.5 (* s s)) x) x))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -200.0f) {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = ((0.25f / s) * x) + 0.5f;
	} else {
		tmp = 1.0f / (((0.5f / (s * s)) * x) * x);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-200.0))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(Float32(0.25) / s) * x) + Float32(0.5));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) / Float32(s * s)) * x) * x));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 99.0%

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

   if -200 < (/.f32 (neg.f32 x) s) < 0.0500000007

   1. Initial program 99.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 99.3

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

   4. Applied rewrites 99.3%

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

      1. rem-square-sqrt N/A

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

      2. pow1/2 N/A

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

      3. pow1/2 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lift-sqrt.f32 N/A

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

      7. pow1/2 N/A

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

      8. pow-pow N/A

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

      9. lift-exp.f32 N/A

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

      10. pow-exp N/A

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

      11. lower-exp.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. metadata-eval 99.6

          \[\leadsto \frac{1}{1 + {\left(e^{\frac{-x}{s} \cdot \color{blue}{0.25}}\right)}^{2} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   6. Applied rewrites 99.6%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{\frac{-x}{s} \cdot 0.25}\right)}^{2}} \cdot \sqrt{e^{\frac{-x}{s}}}} \]

   7. Taylor expanded in x around 0

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

      1. associate-*r/ N/A

         \[\leadsto \frac{1}{2} + \color{blue}{\frac{\frac{1}{4} \cdot x}{s}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot x}{s} + \frac{1}{2}} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{4}}{s} \cdot x} + \frac{1}{2} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot 1}}{s} \cdot x + \frac{1}{2} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{\left(\frac{1}{4} \cdot \frac{1}{s}\right)} \cdot x + \frac{1}{2} \]

      6. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4} \cdot \frac{1}{s}, x, \frac{1}{2}\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{4} \cdot 1}{s}}, x, \frac{1}{2}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\frac{1}{4}}}{s}, x, \frac{1}{2}\right) \]

      9. lower-/.f32 88.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.25}{s}}, x, 0.5\right) \]

   9. Applied rewrites 86.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.25}{s}, x, 0.5\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.9%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 6.5

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

   5. Applied rewrites 6.5%

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

   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 79.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 79.9%

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

Alternative 8: 88.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -200.0)
   (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))
   (/ 1.0 (+ (* (/ (- (/ (* 0.5 x) s) 1.0) s) x) 2.0))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -200.0f) {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	} else {
		tmp = 1.0f / ((((((0.5f * x) / s) - 1.0f) / s) * x) + 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-200.0))
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * x) / s) - Float32(1.0)) / s) * x) + Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 100.0

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

   7. Applied rewrites 100.0%

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

   if -200 < (/.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. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 39.3

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

   5. Applied rewrites 39.2%

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

   7. Applied rewrites 82.4%

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

Alternative 9: 48.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.2

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

   5. Applied rewrites 27.9%

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

   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 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.2%

       \[\leadsto \frac{1}{\frac{2}{x} \cdot x} \]

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   5. Applied rewrites 63.8%

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

Alternative 10: 48.7% accurate, 2.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		tmp = 1.0f / ((2.0f / x) * x);
	} 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) <= (-1.0e0)) then
        tmp = 1.0e0 / ((2.0e0 / x) * x)
    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(-1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) / x) * x));
	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(-1.0))
		tmp = single(1.0) / ((single(2.0) / x) * x);
	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) < -1

   1. Initial program 99.9%

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

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.2

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

   5. Applied rewrites 27.9%

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

   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 5.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.2%

       \[\leadsto \frac{1}{\frac{2}{x} \cdot x} \]

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   5. Applied rewrites 63.8%

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

Alternative 11: 48.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.2%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 63.8

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

   5. Applied rewrites 63.8%

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

Alternative 12: 47.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 0.05000000074505806) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		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 (t_0 <= 0.05000000074505806e0) 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 (t_0 <= Float32(0.05000000074505806))
		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 (t_0 <= single(0.05000000074505806))
		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 (/.f32 (neg.f32 x) s) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 52.7%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sqrt-prod N/A

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

      6. lower-*.f32 N/A

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

      7. lower-sqrt.f32 N/A

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

      8. lower-sqrt.f32 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 43.0

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

   7. Applied rewrites 43.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 43.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.05000000074505806:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% 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 x around 0

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

5. Applied rewrites 35.2%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))