Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 11.7s

Alternatives: 19

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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf(-(x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(-Float32(x / s))))))
end

Derivation

1. Initial program 99.8%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. inv-pow N/A

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

   6. pow-to-exp N/A

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

   7. *-commutative N/A

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

   8. log-pow N/A

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

   9. inv-pow N/A

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

   10. lift-/.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. log-rec N/A

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

   14. lower-neg.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. lower-log1p.f32 99.8

       \[\leadsto e^{-\color{blue}{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}} \]

5. Final simplification 99.8%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \]
6. Add Preprocessing

Alternative 2: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg N/A

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

   2. neg-mul-1 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f32 N/A

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

   5. lower-exp.f32 N/A

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

   6. lower-/.f32 99.7

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

4. Applied egg-rr 99.7%

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

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. prod-exp N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-exp.f32 N/A

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

   12. metadata-eval N/A

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

   13. lift-/.f32 N/A

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

   14. associate-/l/ N/A

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

   15. lower-/.f32 N/A

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

   16. lower-*.f32 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{-\frac{x}{s}} + 1} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (- (/ x s))) 1.0)) 0.0)
   (/
    1.0
    (fma x (/ (* x (fma 0.5 s (* x -0.16666666666666666))) (* s (* s s))) 2.0))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf(-(x / s)) + 1.0f)) <= 0.0f) {
		tmp = 1.0f / fmaf(x, ((x * fmaf(0.5f, s, (x * -0.16666666666666666f))) / (s * (s * s))), 2.0f);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(-Float32(x / s))) + Float32(1.0))) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(x * fma(Float32(0.5), s, Float32(x * Float32(-0.16666666666666666)))) / Float32(s * Float32(s * s))), Float32(2.0)));
	else
		tmp = Float32(0.5);
	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)))) < 0.0

   1. Initial program 100.0%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot \left(s \cdot x\right)}{{s}^{3}}}, 2\right)} \]
   9. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2} + \frac{1}{2} \cdot \left(s \cdot x\right)}{{s}^{3}}}, 2\right)} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \cdot \left(s \cdot x\right)}{{s}^{3}}, 2\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot x} + \frac{1}{2} \cdot \left(s \cdot x\right)}{{s}^{3}}, 2\right)} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(\frac{-1}{6} \cdot x\right) \cdot x + \color{blue}{\left(\frac{1}{2} \cdot s\right) \cdot x}}{{s}^{3}}, 2\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}, 2\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{x \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}, 2\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \color{blue}{\left(\frac{1}{2} \cdot s + \frac{-1}{6} \cdot x\right)}}{{s}^{3}}, 2\right)} \]

      8. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, s, \frac{-1}{6} \cdot x\right)}}{{s}^{3}}, 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}, 2\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}, 2\right)} \]

      11. cube-mult N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{{s}^{2}}}, 2\right)} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot {s}^{2}}}, 2\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

      15. lower-*.f32 92.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

   10. Simplified 92.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{x \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

   if 0.0 < (/.f32 #s(literal 1 binary32) (+.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

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

   5. Simplified 52.2%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{-\frac{x}{s}} + 1} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{x \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{e^{-\frac{x}{s}} + 1} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{-0.16666666666666666 \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ 1.0 (+ (exp (- (/ x s))) 1.0)) 0.0)
   (/ 1.0 (fma x (/ (* -0.16666666666666666 (* x x)) (* s (* s s))) 2.0))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if ((1.0f / (expf(-(x / s)) + 1.0f)) <= 0.0f) {
		tmp = 1.0f / fmaf(x, ((-0.16666666666666666f * (x * x)) / (s * (s * s))), 2.0f);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) / Float32(exp(Float32(-Float32(x / s))) + Float32(1.0))) <= Float32(0.0))
		tmp = Float32(Float32(1.0) / fma(x, Float32(Float32(Float32(-0.16666666666666666) * Float32(x * x)) / Float32(s * Float32(s * s))), Float32(2.0)));
	else
		tmp = Float32(0.5);
	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)))) < 0.0

   1. Initial program 100.0%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{-1}{6} \cdot \frac{{x}^{2}}{{s}^{3}}}, 2\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2}}{{s}^{3}}}, 2\right)} \]

      2. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2}}{{s}^{3}}}, 2\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{{x}^{2} \cdot \frac{-1}{6}}}{{s}^{3}}, 2\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{{x}^{2} \cdot \frac{-1}{6}}}{{s}^{3}}, 2\right)} \]

      5. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}}{{s}^{3}}, 2\right)} \]

      6. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}}{{s}^{3}}, 2\right)} \]

      7. cube-mult N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\color{blue}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{s \cdot \color{blue}{{s}^{2}}}, 2\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{\color{blue}{s \cdot {s}^{2}}}, 2\right)} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \frac{-1}{6}}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

      11. lower-*.f32 92.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot -0.16666666666666666}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

   10. Simplified 92.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\left(x \cdot x\right) \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

   if 0.0 < (/.f32 #s(literal 1 binary32) (+.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

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

   5. Simplified 52.2%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{e^{-\frac{x}{s}} + 1} \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{-0.16666666666666666 \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 9.999999848243207 \cdot 10^{+30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot 2\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 9.999999848243207e+30)
   0.5
   (/ (* s (* s 2.0)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 9.999999848243207e+30f) {
		tmp = 0.5f;
	} else {
		tmp = (s * (s * 2.0f)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp(-(x / s)) <= 9.999999848243207e+30) then
        tmp = 0.5e0
    else
        tmp = (s * (s * 2.0e0)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(9.999999848243207e+30))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * Float32(s * Float32(2.0))) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(9.999999848243207e+30))
		tmp = single(0.5);
	else
		tmp = (s * (s * single(2.0))) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 9.99999985e30

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.2%

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

   if 9.99999985e30 < (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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 79.6

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

   8. Simplified 79.6%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. /-rgt-identity N/A

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

      8. clear-num N/A

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

      9. lift-/.f32 N/A

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

      10. clear-num N/A

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

      11. lift-*.f32 N/A

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

      12. lift-fma.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. times-frac N/A

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

   10. Applied egg-rr 66.1%

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

   11. Taylor expanded in s around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f32 N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

       7. lower-*.f32 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f32 78.5

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

   13. Simplified 78.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 9.999999848243207 \cdot 10^{+30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(s \cdot 2\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 9.999999848243207 \cdot 10^{+30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 9.999999848243207e+30)
   0.5
   (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 9.999999848243207e+30f) {
		tmp = 0.5f;
	} else {
		tmp = (2.0f * (s * s)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp(-(x / s)) <= 9.999999848243207e+30) then
        tmp = 0.5e0
    else
        tmp = (2.0e0 * (s * s)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(9.999999848243207e+30))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(s * s)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(9.999999848243207e+30))
		tmp = single(0.5);
	else
		tmp = (single(2.0) * (s * s)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 9.99999985e30

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.2%

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

   if 9.99999985e30 < (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

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 67.2%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 78.5

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

   8. Simplified 78.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 9.999999848243207 \cdot 10^{+30}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 4.0) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 4.0f) {
		tmp = 0.5f;
	} else {
		tmp = -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 (exp(-(x / s)) <= 4.0e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(4.0))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.8%

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

   if 4 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 63.5%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f32 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f32 30.1

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

   11. Simplified 30.1%

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

       1. distribute-frac-neg2 N/A

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

       2. clear-num N/A

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

       3. distribute-neg-frac N/A

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

       4. metadata-eval N/A

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

       5. lower-/.f32 N/A

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

       6. lower-/.f32 33.1

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

   13. Applied egg-rr 33.1%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (- (/ x s))) 4.0) 0.5 (- (/ s x))))

C

float code(float x, float s) {
	float tmp;
	if (expf(-(x / s)) <= 4.0f) {
		tmp = 0.5f;
	} else {
		tmp = -(s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (exp(-(x / s)) <= 4.0e0) then
        tmp = 0.5e0
    else
        tmp = -(s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(-Float32(x / s))) <= Float32(4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(-Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (exp(-(x / s)) <= single(4.0))
		tmp = single(0.5);
	else
		tmp = -(s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 4

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.8%

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

   if 4 < (exp.f32 (/.f32 (neg.f32 x) s))

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 63.5%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 74.5

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

   8. Simplified 74.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f32 N/A

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

       5. mul-1-neg N/A

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

       6. lower-neg.f32 30.1

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

   11. Simplified 30.1%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-\frac{x}{s}} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;-\frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 10: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 5.0)
   0.5
   (/
    1.0
    (fma
     x
     (fma
      x
      (fma -0.16666666666666666 (/ x (* s (* s s))) (/ 0.5 (* s s)))
      (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf(x, fmaf(-0.16666666666666666f, (x / (s * (s * s))), (0.5f / (s * s))), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(x, fma(Float32(-0.16666666666666666), Float32(x / Float32(s * Float32(s * s))), Float32(Float32(0.5) / Float32(s * s))), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 53.4%

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

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

   1. Initial program 99.6%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 99.5

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 91.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, x \cdot 0.5 - s, -0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 100.0)
   0.5
   (/
    1.0
    (fma
     x
     (/ (fma s (- (* x 0.5) s) (* -0.16666666666666666 (* x x))) (* s (* s s)))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 100.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, (fmaf(s, ((x * 0.5f) - s), (-0.16666666666666666f * (x * x))) / (s * (s * s))), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(100.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(s, Float32(Float32(x * Float32(0.5)) - s), Float32(Float32(-0.16666666666666666) * Float32(x * x))) / Float32(s * Float32(s * s))), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 100

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.2%

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

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

   1. Initial program 100.0%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2} + s \cdot \left(-1 \cdot s + \frac{1}{2} \cdot x\right)}{{s}^{3}}}, 2\right)} \]
   9. Step-by-step derivation

      1. lower-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\frac{-1}{6} \cdot {x}^{2} + s \cdot \left(-1 \cdot s + \frac{1}{2} \cdot x\right)}{{s}^{3}}}, 2\right)} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{s \cdot \left(-1 \cdot s + \frac{1}{2} \cdot x\right) + \frac{-1}{6} \cdot {x}^{2}}}{{s}^{3}}, 2\right)} \]

      3. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\color{blue}{\mathsf{fma}\left(s, -1 \cdot s + \frac{1}{2} \cdot x, \frac{-1}{6} \cdot {x}^{2}\right)}}{{s}^{3}}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot x + -1 \cdot s}, \frac{-1}{6} \cdot {x}^{2}\right)}{{s}^{3}}, 2\right)} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x + \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}, \frac{-1}{6} \cdot {x}^{2}\right)}{{s}^{3}}, 2\right)} \]

      6. unsub-neg N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot x - s}, \frac{-1}{6} \cdot {x}^{2}\right)}{{s}^{3}}, 2\right)} \]

      7. lower--.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot x - s}, \frac{-1}{6} \cdot {x}^{2}\right)}{{s}^{3}}, 2\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \color{blue}{\frac{1}{2} \cdot x} - s, \frac{-1}{6} \cdot {x}^{2}\right)}{{s}^{3}}, 2\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}\right)}{{s}^{3}}, 2\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}\right)}{{s}^{3}}, 2\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)}{{s}^{3}}, 2\right)} \]

      12. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}\right)}{{s}^{3}}, 2\right)} \]

      13. cube-mult N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]

      14. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{{s}^{2}}}, 2\right)} \]

      15. lower-*.f32 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot {s}^{2}}}, 2\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, \frac{1}{2} \cdot x - s, \left(x \cdot x\right) \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

      17. lower-*.f32 92.3

          \[\leadsto \frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, 0.5 \cdot x - s, \left(x \cdot x\right) \cdot -0.16666666666666666\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}, 2\right)} \]

   10. Simplified 92.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{\mathsf{fma}\left(s, 0.5 \cdot x - s, \left(x \cdot x\right) \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(s, x \cdot 0.5 - s, -0.16666666666666666 \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x \cdot x} \cdot \frac{s}{\mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 100.0)
   0.5
   (* (/ (* s s) (* x x)) (/ s (fma x -0.16666666666666666 (* s 0.5))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 100.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) / (x * x)) * (s / fmaf(x, -0.16666666666666666f, (s * 0.5f)));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(100.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) / Float32(x * x)) * Float32(s / fma(x, Float32(-0.16666666666666666), Float32(s * Float32(0.5)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 100

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.2%

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

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

   1. Initial program 100.0%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \left(s \cdot {x}^{2}\right)}{{s}^{3}}}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot s\right) \cdot {x}^{2}}}{{s}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(s \cdot \frac{1}{2}\right)} \cdot {x}^{2}}{{s}^{3}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}}{{s}^{3}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2}} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2} + \color{blue}{\left(s \cdot \frac{1}{2}\right) \cdot {x}^{2}}}{{s}^{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2} + \color{blue}{\left(\frac{1}{2} \cdot s\right)} \cdot {x}^{2}}{{s}^{3}}} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}} \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}{{s}^{3}}} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}{{s}^{3}}} \]

      14. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot s + \frac{-1}{6} \cdot x\right)}}{{s}^{3}}} \]

      15. lower-fma.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, s, \frac{-1}{6} \cdot x\right)}}{{s}^{3}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}} \]

      18. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      20. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      21. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      22. lower-*.f32 79.7

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   10. Simplified 79.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} \cdot s + x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}} \]

       2. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot s + \color{blue}{x \cdot \frac{-1}{6}}\right)}{s \cdot \left(s \cdot s\right)}} \]

       3. lift-fma.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}}{s \cdot \left(s \cdot s\right)}} \]

       4. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

       5. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

       6. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}}{s \cdot \left(s \cdot s\right)}} \]

       7. lift-/.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}}} \]

       8. lift-/.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}}} \]

       9. clear-num N/A

          \[\leadsto \color{blue}{\frac{s \cdot \left(s \cdot s\right)}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       10. lift-*.f32 N/A

           \[\leadsto \frac{\color{blue}{s \cdot \left(s \cdot s\right)}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{\color{blue}{\left(s \cdot s\right) \cdot s}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       12. lift-*.f32 N/A

           \[\leadsto \frac{\left(s \cdot s\right) \cdot s}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       13. times-frac N/A

           \[\leadsto \color{blue}{\frac{s \cdot s}{x \cdot x} \cdot \frac{s}{\mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       14. lower-*.f32 N/A

           \[\leadsto \color{blue}{\frac{s \cdot s}{x \cdot x} \cdot \frac{s}{\mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       15. lower-/.f32 N/A

           \[\leadsto \color{blue}{\frac{s \cdot s}{x \cdot x}} \cdot \frac{s}{\mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       16. lower-/.f32 87.0

           \[\leadsto \frac{s \cdot s}{x \cdot x} \cdot \color{blue}{\frac{s}{\mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}} \]

   12. Applied egg-rr 87.0%

       \[\leadsto \color{blue}{\frac{s \cdot s}{x \cdot x} \cdot \frac{s}{\mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x \cdot x} \cdot \frac{s}{\mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x} \cdot \frac{s}{x \cdot \mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 100.0)
   0.5
   (* (/ (* s s) x) (/ s (* x (fma x -0.16666666666666666 (* s 0.5)))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 100.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) / x) * (s / (x * fmaf(x, -0.16666666666666666f, (s * 0.5f))));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(100.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) / x) * Float32(s / Float32(x * fma(x, Float32(-0.16666666666666666), Float32(s * Float32(0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 100

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.2%

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

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

   1. Initial program 100.0%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 95.8%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in s around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \left(s \cdot {x}^{2}\right)}{{s}^{3}}}} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot s\right) \cdot {x}^{2}}}{{s}^{3}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(s \cdot \frac{1}{2}\right)} \cdot {x}^{2}}{{s}^{3}}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}}{{s}^{3}}} \]

      4. lower-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right) + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      7. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2}} + s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}{{s}^{3}}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2} + \color{blue}{\left(s \cdot \frac{1}{2}\right) \cdot {x}^{2}}}{{s}^{3}}} \]

      9. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\left(\frac{-1}{6} \cdot x\right) \cdot {x}^{2} + \color{blue}{\left(\frac{1}{2} \cdot s\right)} \cdot {x}^{2}}{{s}^{3}}} \]

      10. distribute-rgt-out N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}} \]

      11. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}}{{s}^{3}}} \]

      12. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}{{s}^{3}}} \]

      13. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{-1}{6} \cdot x + \frac{1}{2} \cdot s\right)}{{s}^{3}}} \]

      14. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot s + \frac{-1}{6} \cdot x\right)}}{{s}^{3}}} \]

      15. lower-fma.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, s, \frac{-1}{6} \cdot x\right)}}{{s}^{3}}} \]

      16. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}} \]

      17. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, \color{blue}{x \cdot \frac{-1}{6}}\right)}{{s}^{3}}} \]

      18. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      19. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      20. lower-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      21. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      22. lower-*.f32 79.7

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   10. Simplified 79.7%

       \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(0.5, s, x \cdot -0.16666666666666666\right)}{s \cdot \left(s \cdot s\right)}}} \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} \cdot s + x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}} \]

       2. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \left(\frac{1}{2} \cdot s + \color{blue}{x \cdot \frac{-1}{6}}\right)}{s \cdot \left(s \cdot s\right)}} \]

       3. lift-fma.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}}{s \cdot \left(s \cdot s\right)}} \]

       4. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

       5. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

       6. lift-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}}{s \cdot \left(s \cdot s\right)}} \]

       7. lift-/.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}}} \]

       8. lift-/.f32 N/A

          \[\leadsto \frac{1}{\color{blue}{\frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}{s \cdot \left(s \cdot s\right)}}} \]

       9. clear-num N/A

          \[\leadsto \color{blue}{\frac{s \cdot \left(s \cdot s\right)}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       10. lift-*.f32 N/A

           \[\leadsto \frac{\color{blue}{s \cdot \left(s \cdot s\right)}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       11. *-commutative N/A

           \[\leadsto \frac{\color{blue}{\left(s \cdot s\right) \cdot s}}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       12. lift-*.f32 N/A

           \[\leadsto \frac{\left(s \cdot s\right) \cdot s}{\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

       13. lift-*.f32 N/A

           \[\leadsto \frac{\left(s \cdot s\right) \cdot s}{\color{blue}{\left(x \cdot x\right)} \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)} \]

       14. associate-*l* N/A

           \[\leadsto \frac{\left(s \cdot s\right) \cdot s}{\color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)\right)}} \]

       15. times-frac N/A

           \[\leadsto \color{blue}{\frac{s \cdot s}{x} \cdot \frac{s}{x \cdot \mathsf{fma}\left(\frac{1}{2}, s, x \cdot \frac{-1}{6}\right)}} \]

   12. Applied egg-rr 87.0%

       \[\leadsto \color{blue}{\frac{s \cdot s}{x} \cdot \frac{s}{x \cdot \mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot s}{x} \cdot \frac{s}{x \cdot \mathsf{fma}\left(x, -0.16666666666666666, s \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -5.00000011871114 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;-x \leq 5.000000018137469 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -5.00000011871114e-34)
   0.5
   (if (<= (- x) 5.000000018137469e-16)
     (/ 1.0 (fma x (/ (fma (/ x s) 0.5 -1.0) s) 2.0))
     (/ (* (* s (* s s)) -6.0) (* x (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -5.00000011871114e-34f) {
		tmp = 0.5f;
	} else if (-x <= 5.000000018137469e-16f) {
		tmp = 1.0f / fmaf(x, (fmaf((x / s), 0.5f, -1.0f) / s), 2.0f);
	} else {
		tmp = ((s * (s * s)) * -6.0f) / (x * (x * x));
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-5.00000011871114e-34))
		tmp = Float32(0.5);
	elseif (Float32(-x) <= Float32(5.000000018137469e-16))
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(x / s), Float32(0.5), Float32(-1.0)) / s), Float32(2.0)));
	else
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -5.00000012e-34

   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. Simplified 39.0%

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

   if -5.00000012e-34 < (neg.f32 x) < 5.00000002e-16

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 74.5%

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

      1. lift-/.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f32 N/A

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

      6. lower-/.f32 86.2

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

      7. lift-fma.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f32 86.2

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

   7. Applied egg-rr 86.2%

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

   if 5.00000002e-16 < (neg.f32 x)

   1. Initial program 99.8%

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

      1. lift-neg.f32 N/A

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

      2. clear-num N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f32 N/A

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

      6. lift-neg.f32 N/A

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

      7. distribute-frac-neg2 N/A

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

      8. remove-double-neg N/A

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

      9. lower-/.f32 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   7. Simplified 92.2%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s \cdot \left(s \cdot s\right)}, \frac{0.5}{s \cdot s}\right), \frac{-1}{s}\right), 2\right)}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      2. lower-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      5. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot \left(s \cdot s\right)\right)} \cdot -6}{{x}^{3}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{{s}^{2}}\right) \cdot -6}{{x}^{3}} \]

      7. lower-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot {s}^{2}\right)} \cdot -6}{{x}^{3}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      10. cube-mult N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{{x}^{2}}} \]

      12. lower-*.f32 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot {x}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      14. lower-*.f32 91.8

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   10. Simplified 91.8%

       \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 62.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 25.0) 0.5 (/ 1.0 (* 0.5 (* x (/ x (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 25.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (0.5f * (x * (x / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 25.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (0.5e0 * (x * (x / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(25.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(0.5) * Float32(x * Float32(x / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(25.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(0.5) * (x * (x / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 25

   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. Simplified 52.9%

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

   if 25 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 65.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 81.5

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

   8. Simplified 81.5%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot s}{\mathsf{fma}\left(x, 0.5, -s\right)}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 25.0) 0.5 (/ (/ (* s s) (fma x 0.5 (- s))) x)))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 25.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) / fmaf(x, 0.5f, -s)) / x;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(25.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) / fma(x, Float32(0.5), Float32(-s))) / x);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 25

   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. Simplified 52.9%

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

   if 25 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 65.3%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 77.2

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

   8. Simplified 77.2%

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

      1. lift-*.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. /-rgt-identity N/A

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

      8. clear-num N/A

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

      9. lift-/.f32 N/A

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

      10. clear-num N/A

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

      11. lift-*.f32 N/A

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

      12. lift-fma.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. distribute-rgt-out N/A

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

      18. times-frac N/A

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

   10. Applied egg-rr 64.2%

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

       1. lift-neg.f32 N/A

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

       2. lift-fma.f32 N/A

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

       3. lift-/.f32 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f32 N/A

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

       6. lift-/.f32 N/A

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

       7. associate-*r/ N/A

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

       8. lift-*.f32 N/A

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

       9. lower-/.f32 79.6

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

   12. Applied egg-rr 79.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 25:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{s \cdot s}{\mathsf{fma}\left(x, 0.5, -s\right)}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 800.0) 0.5 (* 2.0 (* s (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 800.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * (s * (s / (x * x)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 800.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (s * (s / (x * x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(800.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(800.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (s * (s / (x * x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 800

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 51.1%

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

   if 800 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. lower-fma.f32 N/A

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

   5. Simplified 70.1%

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

   6. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f32 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f32 82.2

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

   8. Simplified 82.2%

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

   9. Taylor expanded in x around inf

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f32 N/A

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

       5. lower-/.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 68.8

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

   11. Simplified 68.8%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 800:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 49.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 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. Simplified 28.2%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 58.4

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

   5. Simplified 58.4%

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

4. Final simplification 47.4%

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

Alternative 19: 34.6% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 36.6%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))