Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.8s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% 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 99.7%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. log-pow N/A

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

   5. inv-pow N/A

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

   6. exp-lowering-exp.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\log \left(\frac{1}{1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}}\right)\right) \]

   7. log-rec N/A

      \[\leadsto \mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\log \left(1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   8. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\log \left(1 + e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

   9. log1p-define N/A

      \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\left(\mathsf{log1p}\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   10. log1p-lowering-log1p.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\left(e^{\frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right)\right) \]

   11. exp-lowering-exp.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right)\right) \]

   12. distribute-frac-neg N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right)\right) \]

   13. distribute-frac-neg2 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right)\right) \]

   14. /-lowering-/.f32 N/A

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, \left(\mathsf{neg}\left(s\right)\right)\right)\right)\right)\right)\right) \]

   15. neg-lowering-neg.f32 99.7%

       \[\leadsto \mathsf{exp.f32}\left(\mathsf{neg.f32}\left(\mathsf{log1p.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-0.3333333333333333}{\frac{s}{x}}} \cdot e^{-0.6666666666666666 \cdot \frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (exp (/ -0.3333333333333333 (/ s x)))
    (exp (* -0.6666666666666666 (/ x s)))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf((-0.3333333333333333f / (s / x))) * expf((-0.6666666666666666f * (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(((-0.3333333333333333e0) / (s / x))) * exp(((-0.6666666666666666e0) * (x / s)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(exp(Float32(Float32(-0.3333333333333333) / Float32(s / x))) * exp(Float32(Float32(-0.6666666666666666) * Float32(x / s))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp((single(-0.3333333333333333) / (s / x))) * exp((single(-0.6666666666666666) * (x / s)))));
end

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

   7. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

   9. neg-lowering-neg.f32 99.7%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

   1. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

   2. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

   7. frac-2neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.7%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

8. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{e^{\frac{-0.3333333333333333}{\frac{s}{x}}} \cdot e^{0.6666666666666666 \cdot \frac{-x}{s}}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\frac{-1}{3}, \mathsf{/.f32}\left(s, x\right)\right)\right), \mathsf{exp.f32}\left(\color{blue}{\left(\frac{-2}{3} \cdot \frac{x}{s}\right)}\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\frac{-1}{3}, \mathsf{/.f32}\left(s, x\right)\right)\right), \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\frac{-2}{3}, \left(\frac{x}{s}\right)\right)\right)\right)\right)\right) \]

    2. /-lowering-/.f32 99.7%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{exp.f32}\left(\mathsf{/.f32}\left(\frac{-1}{3}, \mathsf{/.f32}\left(s, x\right)\right)\right), \mathsf{exp.f32}\left(\mathsf{*.f32}\left(\frac{-2}{3}, \mathsf{/.f32}\left(x, s\right)\right)\right)\right)\right)\right) \]

11. Simplified 99.7%

    \[\leadsto \frac{1}{1 + e^{\frac{-0.3333333333333333}{\frac{s}{x}}} \cdot e^{\color{blue}{-0.6666666666666666 \cdot \frac{x}{s}}}} \]
12. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(-1.0) / Float32(s / x)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (single(-1.0) / (s / x))));
end

Derivation

1. Initial program 99.7%

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

   1. *-lft-identity N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

   3. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

   7. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

   9. neg-lowering-neg.f32 99.7%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

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

   1. pow-lowering-pow.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

   2. E-lowering-E.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

   6. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

   7. frac-2neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.7%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

Alternative 5: 66.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000063421537e-30)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 s)
       (* x (/ (+ 0.5 (/ (* x -0.16666666666666666) s)) (* s s)))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000063421537e-30f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((0.5f + ((x * -0.16666666666666666f) / s)) / (s * s))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.0000000063421537e-30)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((0.5e0 + ((x * (-0.16666666666666666e0)) / s)) / (s * s))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000063421537e-30))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.16666666666666666)) / s)) / Float32(s * s)))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000063421537e-30))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((single(0.5) + ((x * single(-0.16666666666666666)) / s)) / (s * s))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e-30

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(-1 \cdot \frac{\log \mathsf{E}\left(\right)}{s} + x \cdot \left(\frac{-1}{6} \cdot \frac{x \cdot {\log \mathsf{E}\left(\right)}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right) \]

   7. Simplified 86.0%

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

   8. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}}{{s}^{2}}\right)}\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      2. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot \frac{x}{s}\right)\right), \left({\color{blue}{s}}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{\frac{-1}{6} \cdot x}{s}\right)\right), \left({s}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), s\right)\right), \left({s}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 87.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 87.3%

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

   if -2e-30 < x

   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 49.8%

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

Alternative 6: 65.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.0000000817356035 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{x \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -3.0000000817356035e-27)
   (/
    1.0
    (+
     2.0
     (* x (+ (/ -1.0 s) (* x (/ (* x -0.16666666666666666) (* s (* s s))))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -3.0000000817356035e-27f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((x * -0.16666666666666666f) / (s * (s * s)))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.0000000817356035e-27)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((x * (-0.16666666666666666e0)) / (s * (s * s)))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.0000000817356035e-27))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(x * Float32(-0.16666666666666666)) / Float32(s * Float32(s * s))))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.0000000817356035e-27))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((x * single(-0.16666666666666666)) / (s * (s * s)))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.00000008e-27

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.7%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(-1 \cdot \frac{\log \mathsf{E}\left(\right)}{s} + x \cdot \left(\frac{-1}{6} \cdot \frac{x \cdot {\log \mathsf{E}\left(\right)}^{3}}{{s}^{3}} + \frac{1}{2} \cdot \frac{{\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)\right)}\right) \]

   7. Simplified 87.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}}\right)}\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6} \cdot x}{\color{blue}{{s}^{3}}}\right)\right)\right)\right)\right)\right) \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), \color{blue}{\left({s}^{3}\right)}\right)\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), \left({\color{blue}{s}}^{3}\right)\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left({\color{blue}{s}}^{3}\right)\right)\right)\right)\right)\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot {s}^{\color{blue}{2}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f32 86.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(-1, s\right), \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 86.0%

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

   if -3.00000008e-27 < x

   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 51.4%

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

Alternative 7: 63.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000063421537e-30)
   (/ 1.0 (+ 2.0 (* x (+ (/ -1.0 s) (* x (/ 0.5 (* s s)))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000063421537e-30f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (0.5f / (s * s))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.0000000063421537e-30)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (0.5e0 / (s * s))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000063421537e-30))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(0.5) / Float32(s * s)))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000063421537e-30))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (single(0.5) / (s * s))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e-30

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

      2. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

      9. /-lowering-/.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + \left(-1 \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2} \cdot \frac{{x}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]

   9. Simplified 83.7%

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

   if -2e-30 < x

   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 49.8%

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

Alternative 8: 63.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000063421537 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000063421537e-30)
   (/ 1.0 (+ 2.0 (* x (* x (/ 0.5 (* s s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000063421537e-30f) {
		tmp = 1.0f / (2.0f + (x * (x * (0.5f / (s * s)))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.0000000063421537e-30)) then
        tmp = 1.0e0 / (2.0e0 + (x * (x * (0.5e0 / (s * s)))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000063421537e-30))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(x * Float32(Float32(0.5) / Float32(s * s))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000063421537e-30))
		tmp = single(1.0) / (single(2.0) + (x * (x * (single(0.5) / (s * s)))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2e-30

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

      2. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

      9. /-lowering-/.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + \left(-1 \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2} \cdot \frac{{x}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]

   9. Simplified 83.7%

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

   10. Taylor expanded in s around 0

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{{s}^{2}}\right)}\right)\right)\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2} \cdot x}{\color{blue}{{s}^{2}}}\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{x \cdot \frac{1}{2}}{{\color{blue}{s}}^{2}}\right)\right)\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(x \cdot \color{blue}{\frac{\frac{1}{2}}{{s}^{2}}}\right)\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(x \cdot \frac{\frac{1}{2} \cdot 1}{{\color{blue}{s}}^{2}}\right)\right)\right)\right) \]

       5. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(x \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{{s}^{2}}}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{{s}^{2}}\right)}\right)\right)\right)\right) \]

       7. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2} \cdot 1}{\color{blue}{{s}^{2}}}\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \left(\frac{\frac{1}{2}}{{\color{blue}{s}}^{2}}\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f32 81.7%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   12. Simplified 81.7%

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

   if -2e-30 < x

   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 49.8%

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

Alternative 9: 58.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.000000314464115 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -6.000000314464115e-23) (/ 1.0 (/ (* 0.5 (* x x)) (* s s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -6.000000314464115e-23f) {
		tmp = 1.0f / ((0.5f * (x * x)) / (s * s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-6.000000314464115e-23)) then
        tmp = 1.0e0 / ((0.5e0 * (x * x)) / (s * s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-6.000000314464115e-23))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) * Float32(x * x)) / Float32(s * s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-6.000000314464115e-23))
		tmp = single(1.0) / ((single(0.5) * (x * x)) / (s * s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -6.00000031e-23

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

      2. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

      9. /-lowering-/.f32 99.7%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + \left(-1 \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2} \cdot \frac{{x}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]

   9. Simplified 83.9%

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

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{1}{2} \cdot \frac{{x}^{2}}{{s}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{1}{2} \cdot {x}^{2}}{\color{blue}{{s}^{2}}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{{x}^{2} \cdot \frac{1}{2}}{{\color{blue}{s}}^{2}}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{{x}^{2} \cdot \left(\frac{1}{18} + \frac{4}{9}\right)}{{s}^{2}}\right)\right) \]

       4. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{1}{18} \cdot {x}^{2} + \frac{4}{9} \cdot {x}^{2}}{{\color{blue}{s}}^{2}}\right)\right) \]

       5. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{1}{18} \cdot {x}^{2} + \frac{4}{9} \cdot {x}^{2}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

       6. distribute-rgt-out N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left({x}^{2} \cdot \left(\frac{1}{18} + \frac{4}{9}\right)\right), \left({\color{blue}{s}}^{2}\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{1}{2}\right), \left({s}^{2}\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{1}{2}\right), \left({\color{blue}{s}}^{2}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{1}{2}\right), \left({s}^{2}\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left({s}^{2}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \left(s \cdot \color{blue}{s}\right)\right)\right) \]

       12. *-lowering-*.f32 72.1%

           \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{1}{2}\right), \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right) \]

   12. Simplified 72.1%

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

   if -6.00000031e-23 < x

   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 50.0%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.000000314464115 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{\frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.000000314464115 \cdot 10^{-23}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -6.000000314464115e-23) (/ (* 2.0 (* s s)) (* x x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -6.000000314464115e-23f) {
		tmp = (2.0f * (s * s)) / (x * x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-6.000000314464115e-23)) then
        tmp = (2.0e0 * (s * s)) / (x * x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-6.000000314464115e-23))
		tmp = Float32(Float32(Float32(2.0) * Float32(s * s)) / Float32(x * x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-6.000000314464115e-23))
		tmp = (single(2.0) * (s * s)) / (x * x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -6.00000031e-23

   1. Initial program 99.6%

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

      1. *-lft-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

      2. exp-prod N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

      3. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

      4. exp-1-e N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      5. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

      7. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{x}{\color{blue}{\mathsf{neg}\left(s\right)}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \color{blue}{\left(\mathsf{neg}\left(s\right)\right)}\right)\right)\right)\right) \]

      9. neg-lowering-neg.f32 99.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(x, \mathsf{neg.f32}\left(s\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.6%

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

      1. pow-lowering-pow.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \color{blue}{\left(\frac{x}{\mathsf{neg}\left(s\right)}\right)}\right)\right)\right) \]

      2. E-lowering-E.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{x}}{\mathsf{neg}\left(s\right)}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{1}{\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}}\right)\right)\right)\right) \]

      4. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\frac{\mathsf{neg}\left(s\right)}{x}\right)}}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(s\right)}{x}}\right)}\right)\right)\right)\right) \]

      6. distribute-frac-neg2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{\mathsf{neg}\left(s\right)}{\color{blue}{\mathsf{neg}\left(x\right)}}}\right)\right)\right)\right) \]

      7. frac-2neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{-1}{\frac{s}{\color{blue}{x}}}\right)\right)\right)\right) \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{s}{x}\right)}\right)\right)\right)\right) \]

      9. /-lowering-/.f32 99.7%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + \left(-1 \cdot \frac{x \cdot \log \mathsf{E}\left(\right)}{s} + \frac{1}{2} \cdot \frac{{x}^{2} \cdot {\log \mathsf{E}\left(\right)}^{2}}{{s}^{2}}\right)\right)}\right) \]

   9. Simplified 83.9%

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

   10. Taylor expanded in x around inf

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

       1. associate-*r/ N/A

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

       2. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(2 \cdot {s}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left({s}^{2}\right)\right), \left({\color{blue}{x}}^{2}\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \left(s \cdot s\right)\right), \left({x}^{2}\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(s, s\right)\right), \left({x}^{2}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(s, s\right)\right), \left(x \cdot \color{blue}{x}\right)\right) \]

       7. *-lowering-*.f32 70.6%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right) \]

   12. Simplified 70.6%

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

   if -6.00000031e-23 < x

   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 50.0%

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

Alternative 11: 48.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.000000036005019 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.000000036005019e-35) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.000000036005019e-35f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 2.000000036005019e-35) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(2.000000036005019e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(2.000000036005019e-35))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 2.00000004e-35

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 52.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 52.6%

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

   if 2.00000004e-35 < x

   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 42.9%

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

Alternative 12: 34.8% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

5. Simplified 34.7%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))