Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \frac{e^{\frac{x}{s} \cdot -0.5}}{e^{0.5 \cdot \frac{x}{s}}}} \end{array} \]

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf(((x / s) * -0.5f)) / 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 / s) * (-0.5e0))) / 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 / s) * Float32(-0.5))) / exp(Float32(Float32(0.5) * Float32(x / s))))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. 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. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-/.f32 N/A

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

   16. sqrt-div N/A

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

   17. lower-/.f32 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around inf

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

   1. exp-sqrt-rev N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. lower-/.f32 99.8

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

7. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + \frac{\color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}}{\sqrt{e^{\frac{x}{s}}}}} \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

   \[\leadsto \frac{1}{1 + \frac{e^{\frac{x}{s} \cdot -0.5}}{\sqrt{e^{\frac{x}{s}}}}} \]
10. Step-by-step derivation

    1. lift-sqrt.f32 N/A

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

    2. pow1/2 N/A

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

    3. lift-exp.f32 N/A

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

    4. pow-exp N/A

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

    5. *-commutative N/A

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

    6. lift-/.f32 N/A

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

    7. associate-/l* N/A

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

    8. lift-*.f32 N/A

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

    9. lift-/.f32 N/A

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

    10. lower-exp.f32 99.9

        \[\leadsto \frac{1}{1 + \frac{e^{\frac{x}{s} \cdot -0.5}}{\color{blue}{e^{\frac{0.5 \cdot x}{s}}}}} \]

    11. lift-/.f32 N/A

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

    12. lift-*.f32 N/A

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

    13. associate-/l* N/A

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

    14. lift-/.f32 N/A

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

    15. lower-*.f32 99.9

        \[\leadsto \frac{1}{1 + \frac{e^{\frac{x}{s} \cdot -0.5}}{e^{\color{blue}{0.5 \cdot \frac{x}{s}}}}} \]

11. Applied rewrites 99.9%

    \[\leadsto \frac{1}{1 + \frac{e^{\frac{x}{s} \cdot -0.5}}{\color{blue}{e^{0.5 \cdot \frac{x}{s}}}}} \]
12. Add Preprocessing

Alternative 2: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot 0.5}\\ \mathbf{elif}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 0.05000000074505806)
     (/ 1.0 (* (* x (/ (/ x s) s)) 0.5))
     (if (<= t_0 0.800000011920929)
       (+ 0.5 (* 0.125 (/ (+ x x) s)))
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = 1.0f / ((x * ((x / s) / s)) * 0.5f);
	} else if (t_0 <= 0.800000011920929f) {
		tmp = 0.5f + (0.125f * ((x + x) / s));
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(x / s) / s)) * Float32(0.5)));
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = Float32(Float32(0.5) + Float32(Float32(0.125) * Float32(Float32(x + x) / s)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

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

   8. Applied rewrites 78.5%

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

   if 0.0500000007 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   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. lift-exp.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. distribute-frac-neg N/A

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

      9. exp-neg N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. distribute-frac-neg N/A

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

      15. lift-/.f32 N/A

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

      16. sqrt-div N/A

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

      17. lower-/.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 86.3

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

   7. Applied rewrites 84.8%

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

   9. Applied rewrites 95.4%

      \[\leadsto 0.5 - \color{blue}{0.125 \cdot \frac{\left(-x\right) - x}{s}} \]

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-fma.f32 N/A

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

      4. *-lft-identity 99.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\left(x \cdot \frac{\frac{x}{s}}{s}\right) \cdot 0.5}\\ \mathbf{elif}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \mathbf{elif}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 0.05000000074505806)
     (/ 1.0 (/ (* x (- (* 0.5 x) s)) (* s s)))
     (if (<= t_0 0.800000011920929)
       (+ 0.5 (* 0.125 (/ (+ x x) s)))
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = 1.0f / ((x * ((0.5f * x) - s)) / (s * s));
	} else if (t_0 <= 0.800000011920929f) {
		tmp = 0.5f + (0.125f * ((x + x) / s));
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(Float32(0.5) * x) - s)) / Float32(s * s)));
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = Float32(Float32(0.5) + Float32(Float32(0.125) * Float32(Float32(x + x) / s)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 s around 0

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

   8. Applied rewrites 76.1%

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

   10. Applied rewrites 76.1%

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

   if 0.0500000007 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   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. lift-exp.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. distribute-frac-neg N/A

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

      9. exp-neg N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. distribute-frac-neg N/A

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

      15. lift-/.f32 N/A

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

      16. sqrt-div N/A

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

      17. lower-/.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 86.3

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

   7. Applied rewrites 84.8%

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

   9. Applied rewrites 95.4%

      \[\leadsto 0.5 - \color{blue}{0.125 \cdot \frac{\left(-x\right) - x}{s}} \]

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-fma.f32 N/A

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

      4. *-lft-identity 99.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 \cdot x - s\right)}{s \cdot s}}\\ \mathbf{elif}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{-x}{s}}}\\ \mathbf{if}\;t\_0 \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s}}\\ \mathbf{elif}\;t\_0 \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (+ 1.0 (exp (/ (- x) s))))))
   (if (<= t_0 0.05000000074505806)
     (/ 1.0 (/ (* (* 0.5 x) x) (* s s)))
     (if (<= t_0 0.800000011920929)
       (+ 0.5 (* 0.125 (/ (+ x x) s)))
       (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))))

C

float code(float x, float s) {
	float t_0 = 1.0f / (1.0f + expf((-x / s)));
	float tmp;
	if (t_0 <= 0.05000000074505806f) {
		tmp = 1.0f / (((0.5f * x) * x) / (s * s));
	} else if (t_0 <= 0.800000011920929f) {
		tmp = 0.5f + (0.125f * ((x + x) / s));
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.05000000074505806))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(0.5) * x) * x) / Float32(s * s)));
	elseif (t_0 <= Float32(0.800000011920929))
		tmp = Float32(Float32(0.5) + Float32(Float32(0.125) * Float32(Float32(x + x) / s)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(\frac{1}{2} \cdot \frac{x}{{s}^{2}} - \frac{1}{s}\right)}} \]
   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 s around 0

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 76.1%

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

   if 0.0500000007 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   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. lift-exp.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. distribute-frac-neg N/A

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

      9. exp-neg N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

      12. remove-double-neg N/A

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

      13. lift-neg.f32 N/A

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

      14. distribute-frac-neg N/A

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

      15. lift-/.f32 N/A

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

      16. sqrt-div N/A

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

      17. lower-/.f32 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in s around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

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

      4. lower--.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 86.3

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

   7. Applied rewrites 84.8%

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

   9. Applied rewrites 95.4%

      \[\leadsto 0.5 - \color{blue}{0.125 \cdot \frac{\left(-x\right) - x}{s}} \]

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-fma.f32 N/A

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

      4. *-lft-identity 99.3

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

   9. Applied rewrites 99.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.05000000074505806:\\ \;\;\;\;\frac{1}{\frac{\left(0.5 \cdot x\right) \cdot x}{s \cdot s}}\\ \mathbf{elif}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;0.5 + 0.125 \cdot \frac{x + x}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5 \cdot \frac{x}{s} - 1}{s}, x, 1\right), 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.800000011920929)
   (/ 1.0 (+ (* (/ (- (/ (* 0.5 x) s) 1.0) s) x) 2.0))
   (/ 1.0 (fma (fma (/ (- (* 0.5 (/ x s)) 1.0) s) x 1.0) 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.800000011920929f) {
		tmp = 1.0f / ((((((0.5f * x) / s) - 1.0f) / s) * x) + 2.0f);
	} else {
		tmp = 1.0f / fmaf(fmaf((((0.5f * (x / s)) - 1.0f) / s), x, 1.0f), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * x) / s) - Float32(1.0)) / s) * x) + Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / fma(fma(Float32(Float32(Float32(Float32(0.5) * Float32(x / s)) - Float32(1.0)) / s), x, Float32(1.0)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 84.9%

      \[\leadsto \frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + \color{blue}{2}} \]

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l/ N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f32 N/A

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

      17. lower-/.f32 28.1

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

   5. Applied rewrites 28.1%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 99.3

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

   7. Applied rewrites 99.3%

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

      1. *-rgt-identity N/A

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

      2. *-rgt-identity N/A

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

      3. lower-*.f32 N/A

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

      4. *-rgt-identity 99.3

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

   9. Applied rewrites 99.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.5 \cdot \frac{x}{s} - 1}{s}, x, 1\right), 1, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1 - \frac{x}{s}, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.800000011920929)
   (/ 1.0 (+ (* (/ (- (/ (* 0.5 x) s) 1.0) s) x) 2.0))
   (/ 1.0 (fma (- 1.0 (/ x s)) 1.0 1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.800000011920929f) {
		tmp = 1.0f / ((((((0.5f * x) / s) - 1.0f) / s) * x) + 2.0f);
	} else {
		tmp = 1.0f / fmaf((1.0f - (x / s)), 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.5) * x) / s) - Float32(1.0)) / s) * x) + Float32(2.0)));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(Float32(1.0) - Float32(x / s)), Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 38.3

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

   5. Applied rewrites 38.3%

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

   7. Applied rewrites 84.9%

      \[\leadsto \frac{1}{\frac{\frac{0.5 \cdot x}{s} - 1}{s} \cdot x + \color{blue}{2}} \]

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-fma.f32 N/A

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

      4. *-lft-identity 99.3

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

   9. Applied rewrites 99.3%

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

Alternative 7: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(t\_0, 1, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ x s))))
   (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.800000011920929)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (fma t_0 1.0 1.0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (x / s);
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.800000011920929f) {
		tmp = 1.0f / (1.0f + t_0);
	} else {
		tmp = 1.0f / fmaf(t_0, 1.0f, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0));
	else
		tmp = Float32(Float32(1.0) / fma(t_0, Float32(1.0), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

      1. lift-fma.f32 N/A

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

      2. *-rgt-identity N/A

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

      3. lower-fma.f32 N/A

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

      4. *-lft-identity 99.3

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

   9. Applied rewrites 98.3%

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

Alternative 8: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{x}{s}\\ \mathbf{if}\;\frac{1}{1 + e^{\frac{-x}{s}}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- 1.0 (/ x s))))
   (if (<= (/ 1.0 (+ 1.0 (exp (/ (- x) s)))) 0.800000011920929)
     (/ 1.0 (+ 1.0 t_0))
     (/ 1.0 (fma 1.0 t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = 1.0f - (x / s);
	float tmp;
	if ((1.0f / (1.0f + expf((-x / s)))) <= 0.800000011920929f) {
		tmp = 1.0f / (1.0f + t_0);
	} else {
		tmp = 1.0f / fmaf(1.0f, t_0, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) - Float32(x / s))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(-x) / s)))) <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + t_0));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), t_0, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

Alternative 9: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;\frac{1}{1 + e^{t\_0}} \leq 0.800000011920929:\\ \;\;\;\;\frac{1}{1 + \left(1 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(1, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= (/ 1.0 (+ 1.0 (exp t_0))) 0.800000011920929)
     (/ 1.0 (+ 1.0 (- 1.0 (/ x s))))
     (/ 1.0 (fma 1.0 t_0 1.0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if ((1.0f / (1.0f + expf(t_0))) <= 0.800000011920929f) {
		tmp = 1.0f / (1.0f + (1.0f - (x / s)));
	} else {
		tmp = 1.0f / fmaf(1.0f, t_0, 1.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(Float32(1.0) + exp(t_0))) <= Float32(0.800000011920929))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / fma(Float32(1.0), t_0, Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))) < 0.800000012

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   if 0.800000012 < (/.f32 #s(literal 1 binary32) (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))))

   1. Initial program 100.0%

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

   3. 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 4.9

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

   5. Applied rewrites 4.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 - \frac{x}{s}\right)}} \]
   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 99.3

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 98.5%

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

Alternative 10: 49.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5) 0.5 (/ 1.0 (+ 1.0 (- 1.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 1.5e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(1.5))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   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 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + -1 \cdot \frac{x}{s}\right)}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 11: 49.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 1.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (+ 1.0 (exp (/ (- x) s))) 1.5) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 1.5f) {
		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 ((1.0e0 + exp((-x / s))) <= 1.5e0) 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(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(1.5))
		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 ((single(1.0) + exp((-x / s))) <= single(1.5))
		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 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 1.5

   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 1.5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

Alternative 12: 48.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;1 + e^{t\_0} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= (+ 1.0 (exp t_0)) 5.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if ((1.0f + expf(t_0)) <= 5.0f) {
		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 ((1.0e0 + exp(t_0)) <= 5.0e0) 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 (Float32(Float32(1.0) + exp(t_0)) <= Float32(5.0))
		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 ((single(1.0) + exp(t_0)) <= single(5.0))
		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 (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 5

   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 5 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

   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 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 s around 0

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

   8. Applied rewrites 76.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.7%

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

Alternative 13: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

Alternative 14: 14.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.25, x, 0.5 \cdot s\right)}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (fma 0.25 x (* 0.5 s)) s))

C

float code(float x, float s) {
	return fmaf(0.25f, x, (0.5f * s)) / s;
}

Julia

function code(x, s)
	return Float32(fma(Float32(0.25), x, Float32(Float32(0.5) * s)) / s)
end

Derivation

1. Initial program 99.8%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. sqrt-pow1 N/A

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

   4. pow2 N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. remove-double-neg N/A

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

   13. lift-neg.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. lift-/.f32 N/A

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

   16. sqrt-div N/A

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

   17. lower-/.f32 N/A

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in s around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 34.6

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

7. Applied rewrites 34.7%

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

8. Taylor expanded in s around 0

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

10. Applied rewrites 34.6%

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

11. Final simplification 34.6%

    \[\leadsto \frac{\mathsf{fma}\left(0.25, x, 0.5 \cdot s\right)}{s} \]
12. Add Preprocessing

Alternative 15: 35.3% 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 34.6%

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

Reproduce

herbie shell --seed 2024339 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))