Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 10.6s

Alternatives: 17

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

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

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 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-1.0)) ^ (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. distribute-frac-neg N/A

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

   2. neg-mul-1 N/A

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

   3. exp-prod N/A

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

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

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

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

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

   6. /-lowering-/.f32 99.7%

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

4. Applied egg-rr 99.7%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(((float) M_E), (0.0f - (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(0.0) - Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (single(0.0) - (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. /-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. Final simplification 99.7%

   \[\leadsto \frac{1}{1 + {e}^{\left(0 - \frac{x}{s}\right)}} \]
6. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (expf((0.0f - (x / s))) + 1.0f);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (exp((0.0e0 - (x / s))) + 1.0e0)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 5: 66.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{1}{s}}{\frac{1}{x \cdot \left(-1 + x \cdot \frac{0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}}{s}\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -20.0)
   0.5
   (/
    1.0
    (+
     2.0
     (/
      (/ 1.0 s)
      (/
       1.0
       (*
        x
        (+ -1.0 (* x (/ (+ 0.5 (/ x (/ s -0.16666666666666666))) s))))))))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((1.0f / s) / (1.0f / (x * (-1.0f + (x * ((0.5f + (x / (s / -0.16666666666666666f))) / s)))))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((1.0e0 / s) / (1.0e0 / (x * ((-1.0e0) + (x * ((0.5e0 + (x / (s / (-0.16666666666666666e0)))) / s)))))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(1.0) / s) / Float32(Float32(1.0) / Float32(x * Float32(Float32(-1.0) + Float32(x * Float32(Float32(Float32(0.5) + Float32(x / Float32(s / Float32(-0.16666666666666666)))) / s))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + ((single(1.0) / s) / (single(1.0) / (x * (single(-1.0) + (x * ((single(0.5) + (x / (s / single(-0.16666666666666666)))) / s)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 83.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\frac{x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s} + x \cdot \left(x \cdot 0.5\right)}{s} - x}{s}}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l* N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

      13. /-lowering-/.f32 N/A

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

      14. *-lowering-*.f32 84.7%

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

   7. Applied egg-rr 84.7%

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{1}{s} \cdot \left(x \cdot \left(x \cdot \left(0.5 + \frac{x \cdot -0.16666666666666666}{s}\right)\right)\right)} - x}{s}} \]
   8. Step-by-step derivation

      1. clear-num N/A

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

      2. div-inv N/A

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

      3. associate-/r* N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. neg-mul-1 N/A

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

      11. distribute-rgt-out N/A

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

      12. associate-*r* N/A

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

   9. Applied egg-rr 87.9%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{1}{s}}{\frac{1}{x \cdot \left(-1 + x \cdot \frac{0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}}{s}\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}\right)}{\frac{s}{\frac{x}{s}}} - \left(\frac{x}{s} + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -20.0)
   0.5
   (/
    1.0
    (-
     (/ (* x (+ 0.5 (/ x (/ s -0.16666666666666666)))) (/ s (/ x s)))
     (+ (/ x s) -2.0)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (((x * (0.5f + (x / (s / -0.16666666666666666f)))) / (s / (x / s))) - ((x / s) + -2.0f));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((x * (0.5e0 + (x / (s / (-0.16666666666666666e0))))) / (s / (x / s))) - ((x / s) + (-2.0e0)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(x * Float32(Float32(0.5) + Float32(x / Float32(s / Float32(-0.16666666666666666))))) / Float32(s / Float32(x / s))) - Float32(Float32(x / s) + Float32(-2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (((x * (single(0.5) + (x / (s / single(-0.16666666666666666))))) / (s / (x / s))) - ((x / s) + single(-2.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 83.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\frac{x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s} + x \cdot \left(x \cdot 0.5\right)}{s} - x}{s}}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l* N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

      13. /-lowering-/.f32 N/A

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

      14. *-lowering-*.f32 84.7%

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

   7. Applied egg-rr 84.7%

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{1}{s} \cdot \left(x \cdot \left(x \cdot \left(0.5 + \frac{x \cdot -0.16666666666666666}{s}\right)\right)\right)} - x}{s}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. associate-+l- N/A

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

      4. sub-neg N/A

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

      5. remove-double-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. sub-neg N/A

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

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

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

   9. Applied egg-rr 86.3%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}\right)}{\frac{s}{\frac{x}{s}}} - \left(\frac{x}{s} + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{x \cdot \left(0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}\right)}{\frac{s}{x}} - x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -20.0)
   0.5
   (/
    1.0
    (+
     2.0
     (/ (- (/ (* x (+ 0.5 (/ x (/ s -0.16666666666666666)))) (/ s x)) x) s)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((((x * (0.5f + (x / (s / -0.16666666666666666f)))) / (s / x)) - x) / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((((x * (0.5e0 + (x / (s / (-0.16666666666666666e0))))) / (s / x)) - x) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(Float32(x * Float32(Float32(0.5) + Float32(x / Float32(s / Float32(-0.16666666666666666))))) / Float32(s / x)) - x) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + ((((x * (single(0.5) + (x / (s / single(-0.16666666666666666))))) / (s / x)) - x) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 83.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\frac{x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s} + x \cdot \left(x \cdot 0.5\right)}{s} - x}{s}}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l* N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

      13. /-lowering-/.f32 N/A

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

      14. *-lowering-*.f32 84.7%

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

   7. Applied egg-rr 84.7%

      \[\leadsto \frac{1}{2 + \frac{\color{blue}{\frac{1}{s} \cdot \left(x \cdot \left(x \cdot \left(0.5 + \frac{x \cdot -0.16666666666666666}{s}\right)\right)\right)} - x}{s}} \]
   8. Step-by-step derivation

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

      15. /-lowering-/.f32 N/A

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

      16. /-lowering-/.f32 85.9%

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

   9. Applied egg-rr 85.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{x \cdot \left(0.5 + \frac{x}{\frac{s}{-0.16666666666666666}}\right)}{\frac{s}{x}} - x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot \frac{x \cdot \left(0.5 + \frac{x \cdot -0.16666666666666666}{s}\right)}{s} - x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -20.0)
   0.5
   (/
    1.0
    (+
     2.0
     (/ (- (* x (/ (* x (+ 0.5 (/ (* x -0.16666666666666666) s))) s)) x) s)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (((x * ((x * (0.5f + ((x * -0.16666666666666666f) / s))) / s)) - x) / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (((x * ((x * (0.5e0 + ((x * (-0.16666666666666666e0)) / s))) / s)) - x) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(x * Float32(Float32(x * Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.16666666666666666)) / s))) / s)) - x) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (((x * ((x * (single(0.5) + ((x * single(-0.16666666666666666)) / s))) / s)) - x) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 83.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\frac{x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s} + x \cdot \left(x \cdot 0.5\right)}{s} - x}{s}}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

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

      2. distribute-lft-out N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-out N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f32 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{x \cdot \frac{x \cdot \left(0.5 + \frac{x \cdot -0.16666666666666666}{s}\right)}{s} - x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + \frac{\frac{\frac{x}{2}}{s}}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -20.0)
   0.5
   (/ 1.0 (+ 2.0 (* x (+ (/ -1.0 s) (/ (/ (/ x 2.0) s) s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= -20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (((x / 2.0f) / s) / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-20.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (((x / 2.0e0) / s) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(-20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(Float32(Float32(x / Float32(2.0)) / s) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(-20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (((x / single(2.0)) / s) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

   if -20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   5. Simplified 83.9%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. +-commutative N/A

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

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

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

      11. distribute-neg-frac N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f32 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. associate-*r/ N/A

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

      17. *-commutative N/A

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

      18. unpow2 N/A

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

      19. associate-/r* N/A

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

   8. Simplified 83.9%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + \frac{\frac{\frac{x}{2}}{s}}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \frac{x \cdot x}{s}\right)}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) 5000000.0)
   0.5
   (/ 1.0 (/ (* -0.16666666666666666 (* x (/ (* x x) s))) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= 5000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((-0.16666666666666666f * (x * ((x * x) / s))) / (s * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= 5000000.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (((-0.16666666666666666e0) * (x * ((x * x) / s))) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(5000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-0.16666666666666666) * Float32(x * Float32(Float32(x * x) / s))) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(5000000.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(-0.16666666666666666) * (x * ((x * x) / s))) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 47.6%

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

   if 5e6 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 89.7%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\frac{x \cdot \left(x \cdot \left(x \cdot -0.16666666666666666\right)\right)}{s} + x \cdot \left(x \cdot 0.5\right)}{s} - x}{s}}} \]
   6. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l* N/A

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

      6. distribute-lft-out N/A

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

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

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

      8. +-commutative N/A

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

      9. associate-/l* N/A

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

      10. distribute-lft-out N/A

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

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

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

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

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

      13. /-lowering-/.f32 N/A

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

      14. *-lowering-*.f32 90.8%

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

   7. Applied egg-rr 90.8%

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

   8. Taylor expanded in s around 0

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. associate-*r/ N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. /-lowering-/.f32 N/A

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f32 92.6%

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

   10. Simplified 92.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \frac{x \cdot x}{s}\right)}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) 50.0) 0.5 (/ (* -2.0 (/ (* s s) x)) x)))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= 50.0f) {
		tmp = 0.5f;
	} else {
		tmp = (-2.0f * ((s * s) / x)) / x;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= 50.0e0) then
        tmp = 0.5e0
    else
        tmp = ((-2.0e0) * ((s * s) / x)) / x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(50.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(-2.0) * Float32(Float32(s * s) / x)) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(50.0))
		tmp = single(0.5);
	else
		tmp = (single(-2.0) * ((s * s) / x)) / x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 50

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.6%

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

   if 50 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   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 45.5%

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

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

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. /-lowering-/.f32 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in s around inf

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f32 81.9%

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

   11. Simplified 81.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 50:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{s \cdot s}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) 5000000.0) 0.5 (/ 2.0 (/ (* x x) (* s s)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= 5000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f / ((x * x) / (s * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= 5000000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 / ((x * x) / (s * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(5000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) / Float32(Float32(x * x) / Float32(s * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(5000000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) / ((x * x) / (s * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 47.6%

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

   if 5e6 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   5. Simplified 89.7%

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

   6. Taylor expanded in s around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-+l+ N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f32 N/A

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

      16. /-lowering-/.f32 86.0%

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

   8. Simplified 86.0%

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

   9. Taylor expanded in x around inf

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

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. associate-*l/ N/A

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

       10. *-lft-identity N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f32 87.1%

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

   11. Simplified 87.1%

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

       1. div-inv N/A

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

       2. associate-*r/ N/A

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

       3. associate-*l/ N/A

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

       4. div-inv N/A

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

       5. div-inv N/A

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

       6. metadata-eval N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. metadata-eval N/A

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

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

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

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

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

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

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

       13. *-lowering-*.f32 87.1%

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

   13. Applied egg-rr 87.1%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{x \cdot x}{s \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) 5000000.0) 0.5 (* (* s s) (/ 2.0 (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= 5000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * s) * (2.0f / (x * x));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= 5000000.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * s) * (2.0e0 / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(5000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * s) * Float32(Float32(2.0) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(5000000.0))
		tmp = single(0.5);
	else
		tmp = (s * s) * (single(2.0) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 47.6%

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

   if 5e6 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

   5. Simplified 89.7%

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

   6. Taylor expanded in s around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-+l+ N/A

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

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

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

      6. associate-*r/ N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f32 N/A

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

      16. /-lowering-/.f32 86.0%

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

   8. Simplified 86.0%

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

   9. Taylor expanded in x around inf

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

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. metadata-eval N/A

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

       9. associate-*l/ N/A

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

       10. *-lft-identity N/A

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

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

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

       12. unpow2 N/A

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

       13. *-lowering-*.f32 87.1%

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

   11. Simplified 87.1%

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

       1. associate-/r/ N/A

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

       2. associate-*r/ N/A

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

       3. clear-num N/A

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

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

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

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

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

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

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

       7. *-lowering-*.f32 83.6%

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

   13. Applied egg-rr 83.6%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 5000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) -1.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= (-1.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   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 61.5%

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

   5. Simplified 61.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 0.4000000059604645:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- 0.0 (/ x s)) 0.4000000059604645) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if ((0.0f - (x / s)) <= 0.4000000059604645f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((0.0e0 - (x / s)) <= 0.4000000059604645e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(0.0) - Float32(x / s)) <= Float32(0.4000000059604645))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(0.0) - (x / s)) <= single(0.4000000059604645))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.400000006

   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 51.3%

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

   if 0.400000006 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   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 44.6%

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

   5. Simplified 44.6%

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

   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 39.9%

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

   8. Simplified 39.9%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. clear-num N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

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

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

      6. /-lowering-/.f32 44.6%

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

   10. Applied egg-rr 44.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;0 - \frac{x}{s} \leq 0.4000000059604645:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 46.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.0000000467443897e-7) (- (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.0000000467443897e-7f) {
		tmp = -(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 <= (-4.0000000467443897e-7)) then
        tmp = -(s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.0000000467443897e-7))
		tmp = Float32(-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(-4.0000000467443897e-7))
		tmp = -(s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.00000005e-7

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 54.3%

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

   5. Simplified 54.3%

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

   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 48.3%

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

   8. Simplified 48.3%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-neg-frac N/A

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

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

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

      4. neg-lowering-neg.f32 48.3%

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

   10. Applied egg-rr 48.3%

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

   if -4.00000005e-7 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 45.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.4% 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 32.6%

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

Reproduce

herbie shell --seed 2024159 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))