Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.6s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{s}}{-2}\\ \frac{1}{1 + {\left({e}^{0.6666666666666666} \cdot {e}^{0.6666666666666666}\right)}^{t\_0} \cdot {\left(e^{0.3333333333333333}\right)}^{\left(t\_0 \cdot 2\right)}} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (/ x s) -2.0)))
   (/
    1.0
    (+
     1.0
     (*
      (pow (* (pow E 0.6666666666666666) (pow E 0.6666666666666666)) t_0)
      (pow (exp 0.3333333333333333) (* t_0 2.0)))))))

C

float code(float x, float s) {
	float t_0 = (x / s) / -2.0f;
	return 1.0f / (1.0f + (powf((powf(((float) M_E), 0.6666666666666666f) * powf(((float) M_E), 0.6666666666666666f)), t_0) * powf(expf(0.3333333333333333f), (t_0 * 2.0f))));
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) / Float32(-2.0))
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32((Float32((Float32(exp(1)) ^ Float32(0.6666666666666666)) * (Float32(exp(1)) ^ Float32(0.6666666666666666))) ^ t_0) * (exp(Float32(0.3333333333333333)) ^ Float32(t_0 * Float32(2.0))))))
end

MATLAB

function tmp = code(x, s)
	t_0 = (x / s) / single(-2.0);
	tmp = single(1.0) / (single(1.0) + ((((single(2.71828182845904523536) ^ single(0.6666666666666666)) * (single(2.71828182845904523536) ^ single(0.6666666666666666))) ^ t_0) * (exp(single(0.3333333333333333)) ^ (t_0 * single(2.0)))));
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. /-lowering-/.f32 N/A

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

   7. neg-lowering-neg.f32 99.7%

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

4. Applied egg-rr 99.7%

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

   1. sqr-pow N/A

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

   2. pow-prod-down N/A

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

   3. add-cube-cbrt N/A

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

   4. add-cube-cbrt N/A

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

   5. swap-sqr N/A

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

   6. unpow-prod-down N/A

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

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

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

6. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left({e}^{0.6666666666666666} \cdot {e}^{0.6666666666666666}\right)}^{\left(\frac{\frac{x}{s}}{-2}\right)} \cdot {\left(e^{0.3333333333333333}\right)}^{\left(2 \cdot \frac{\frac{x}{s}}{-2}\right)}}} \]

7. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + {\left({e}^{0.6666666666666666} \cdot {e}^{0.6666666666666666}\right)}^{\left(\frac{\frac{x}{s}}{-2}\right)} \cdot {\left(e^{0.3333333333333333}\right)}^{\left(\frac{\frac{x}{s}}{-2} \cdot 2\right)}} \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{0 - \frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- 0.0 (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((0.0f - (x / s)))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(0.0) - 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. /-lowering-/.f32 N/A

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

   13. neg-lowering-neg.f32 99.8%

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{0 - \frac{x}{s}}\right)} \]
6. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{0.6666666666666666 \cdot \frac{x}{s}}\right)}^{-1.5}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp (* 0.6666666666666666 (/ x s))) -1.5))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf((0.6666666666666666f * (x / s))), -1.5f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((0.6666666666666666e0 * (x / s))) ** (-1.5e0)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(Float32(0.6666666666666666) * Float32(x / s))) ^ Float32(-1.5))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp((single(0.6666666666666666) * (x / s))) ^ single(-1.5)));
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. /-lowering-/.f32 N/A

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

   7. neg-lowering-neg.f32 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

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

   3. pow-pow N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right), \color{blue}{\left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E}\left(\right), \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E}\left(\right), \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right)\right) \]

   12. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right)\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \left(0 - \color{blue}{\frac{x}{s}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{x}{s}\right)}\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.7%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e \cdot \left(e \cdot e\right)\right)}^{\left(0.3333333333333333 \cdot \left(0 - \frac{x}{s}\right)\right)}}} \]

8. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{0.6666666666666666 \cdot \frac{x}{s}}\right)}^{-1.5}}} \]
9. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e \cdot \left(e \cdot e\right)\right)}^{\left(\frac{x}{s} \cdot -0.3333333333333333\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (* E (* E E)) (* (/ x s) -0.3333333333333333)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf((((float) M_E) * (((float) M_E) * ((float) M_E))), ((x / s) * -0.3333333333333333f)));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(Float32(exp(1)) * Float32(Float32(exp(1)) * Float32(exp(1)))) ^ Float32(Float32(x / s) * Float32(-0.3333333333333333)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((single(2.71828182845904523536) * (single(2.71828182845904523536) * single(2.71828182845904523536))) ^ ((x / s) * single(-0.3333333333333333))));
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. /-lowering-/.f32 N/A

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

   7. neg-lowering-neg.f32 99.7%

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

4. Applied egg-rr 99.7%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

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

   3. pow-pow N/A

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

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right), \color{blue}{\left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E}\left(\right), \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\color{blue}{\frac{1}{3}} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E}\left(\right), \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \left(\frac{1}{3} \cdot \frac{\mathsf{neg}\left(x\right)}{s}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right)\right) \]

   12. distribute-frac-neg N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right)\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \left(0 - \color{blue}{\frac{x}{s}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{x}{s}\right)}\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.7%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{*.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{E.f32}\left(\right)\right)\right), \mathsf{*.f32}\left(\frac{1}{3}, \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e \cdot \left(e \cdot e\right)\right)}^{\left(0.3333333333333333 \cdot \left(0 - \frac{x}{s}\right)\right)}}} \]

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

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

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

   3. /-lowering-/.f32 99.7%

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

9. Simplified 99.7%

   \[\leadsto \frac{1}{1 + {\left(e \cdot \left(e \cdot e\right)\right)}^{\color{blue}{\left(\frac{x}{s} \cdot -0.3333333333333333\right)}}} \]
10. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{0 - \frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (- 0.0 (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((0.0f - (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((0.0e0 - (x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((single(0.0) - (x / s))));
end

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

   \[\leadsto \frac{1}{1 + e^{0 - \frac{x}{s}}} \]
4. Add Preprocessing

Alternative 6: 66.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s} + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000136226006e-28)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (/ (+ 0.5 (/ (* x -0.16666666666666666) s)) (* s s)))
       (/ -1.0 s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000136226006e-28f) {
		tmp = 1.0f / (2.0f + (x * ((x * ((0.5f + ((x * -0.16666666666666666f) / s)) / (s * s))) + (-1.0f / 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 <= (-5.000000136226006e-28)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * ((0.5e0 + ((x * (-0.16666666666666666e0)) / s)) / (s * s))) + ((-1.0e0) / s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000136226006e-28))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.16666666666666666)) / s)) / Float32(s * s))) + Float32(Float32(-1.0) / s)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000136226006e-28))
		tmp = single(1.0) / (single(2.0) + (x * ((x * ((single(0.5) + ((x * single(-0.16666666666666666)) / s)) / (s * s))) + (single(-1.0) / s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000014e-28

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in s around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

      5. *-commutative N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f32 93.1%

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

   8. Simplified 93.1%

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

   if -5.00000014e-28 < 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 52.5%

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

Alternative 7: 65.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{x \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000136226006e-28)
   (/
    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 <= -5.000000136226006e-28f) {
		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 <= (-5.000000136226006e-28)) 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(-5.000000136226006e-28))
		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(-5.000000136226006e-28))
		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 < -5.00000014e-28

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. cube-mult N/A

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f32 91.9%

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

   8. Simplified 91.9%

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

   if -5.00000014e-28 < 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 52.5%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{x \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.0000000126843074e-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 <= -4.0000000126843074e-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 <= (-4.0000000126843074e-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(-4.0000000126843074e-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(-4.0000000126843074e-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 < -4e-30

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 92.5%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 87.0%

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

   8. Simplified 87.0%

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

   if -4e-30 < 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 52.2%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.0000000126843074 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000136226006e-28)
   (/ 1.0 (+ 2.0 (* x (* x (/ 0.5 (* s s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000136226006e-28f) {
		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 <= (-5.000000136226006e-28)) 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(-5.000000136226006e-28))
		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(-5.000000136226006e-28))
		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 < -5.00000014e-28

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 93.1%

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

   6. Taylor expanded in x around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f32 87.5%

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

   8. Simplified 87.5%

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

   9. Taylor expanded in x around inf

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

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

       3. metadata-eval N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       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 86.7%

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

   11. Simplified 86.7%

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

   if -5.00000014e-28 < 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 52.5%

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

Alternative 10: 61.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;\frac{s \cdot s}{x \cdot -0.16666666666666666} \cdot \frac{s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.00000012549758e-22)
   (* (/ (* s s) (* x -0.16666666666666666)) (/ s (* x x)))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.00000012549758e-22f) {
		tmp = ((s * s) / (x * -0.16666666666666666f)) * (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 <= (-4.00000012549758e-22)) then
        tmp = ((s * s) / (x * (-0.16666666666666666e0))) * (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(-4.00000012549758e-22))
		tmp = Float32(Float32(Float32(s * s) / Float32(x * Float32(-0.16666666666666666))) * Float32(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(-4.00000012549758e-22))
		tmp = ((s * s) / (x * single(-0.16666666666666666))) * (s / (x * x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.00000013e-22

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 93.3%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 78.2%

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

   8. Simplified 78.2%

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

      1. clear-num N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. times-frac N/A

         \[\leadsto \frac{s \cdot s}{\frac{-1}{6} \cdot x} \cdot \color{blue}{\frac{s}{x \cdot x}} \]

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. *-lowering-*.f32 83.3%

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

   10. Applied egg-rr 83.3%

       \[\leadsto \color{blue}{\frac{s \cdot s}{x \cdot -0.16666666666666666} \cdot \frac{s}{x \cdot x}} \]

   if -4.00000013e-22 < 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 52.9%

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

Alternative 11: 61.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.499999956129175 \cdot 10^{-15}:\\ \;\;\;\;s \cdot \frac{s \cdot s}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.499999956129175e-15)
   (* s (/ (* s s) (* -0.16666666666666666 (* x (* x x)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.499999956129175e-15f) {
		tmp = s * ((s * s) / (-0.16666666666666666f * (x * (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 <= (-2.499999956129175e-15)) then
        tmp = s * ((s * s) / ((-0.16666666666666666e0) * (x * (x * x))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.499999956129175e-15))
		tmp = Float32(s * Float32(Float32(s * s) / Float32(Float32(-0.16666666666666666) * Float32(x * 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(-2.499999956129175e-15))
		tmp = s * ((s * s) / (single(-0.16666666666666666) * (x * (x * x))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.49999996e-15

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 92.5%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

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

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f32 88.2%

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

   8. Simplified 88.2%

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

      1. clear-num N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f32 85.2%

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

   10. Applied egg-rr 85.2%

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

   if -2.49999996e-15 < 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 51.6%

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

Alternative 12: 59.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\frac{0.5 \cdot \left(x \cdot x\right)}{s \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.00000012549758e-22) (/ 1.0 (/ (* 0.5 (* x x)) (* s s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.00000012549758e-22f) {
		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 <= (-4.00000012549758e-22)) 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(-4.00000012549758e-22))
		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(-4.00000012549758e-22))
		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 < -4.00000013e-22

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.8%

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

   6. Taylor expanded in x around inf

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 79.3%

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

   8. Simplified 79.3%

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

   if -4.00000013e-22 < 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 52.9%

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

Alternative 13: 58.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.00000012549758 \cdot 10^{-22}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.00000012549758e-22) (/ (* 2.0 (* s s)) (* x x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.00000012549758e-22f) {
		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 <= (-4.00000012549758e-22)) 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(-4.00000012549758e-22))
		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(-4.00000012549758e-22))
		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 < -4.00000013e-22

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 75.8%

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

   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

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

      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 79.1%

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

   8. Simplified 79.1%

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

   if -4.00000013e-22 < 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 52.9%

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

Alternative 14: 49.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.000000093402204 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 2.000000093402204e-34) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 2.000000093402204e-34f) {
		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 <= 2.000000093402204e-34) 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(2.000000093402204e-34))
		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(2.000000093402204e-34))
		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 < 2.00000009e-34

   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 56.9%

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

   5. Simplified 56.9%

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

   if 2.00000009e-34 < x

   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 43.5%

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

Alternative 15: 46.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;0 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.000000026702864e-10) (- 0.0 (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.000000026702864e-10f) {
		tmp = 0.0f - (s / 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 <= (-2.000000026702864e-10)) then
        tmp = 0.0e0 - (s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.000000026702864e-10))
		tmp = Float32(Float32(0.0) - Float32(s / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.000000026702864e-10))
		tmp = single(0.0) - (s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000003e-10

   1. Initial program 99.6%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 50.1%

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

   5. Simplified 50.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

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

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 45.6%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 45.6%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]

   if -2.00000003e-10 < 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 50.9%

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

Alternative 16: 35.5% 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 36.8%

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

Reproduce

herbie shell --seed 2024288 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))