Logistic function

Percentage Accurate: 99.8% → 100.0%

Time: 13.8s

Alternatives: 14

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: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \langle \left( \frac{1}{1 + e^{\frac{x}{-s}}} \right)_{\text{binary64}} \rangle_{\text{binary32}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (cast (! :precision binary64 (/ 1.0 (+ 1.0 (exp (/ x (- s))))))))

C

float code(float x, float s) {
	double tmp = 1.0 / (1.0 + exp((((double) x) / -((double) s))));
	return (float) tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(8) :: tmp
    tmp = 1.0d0 / (1.0d0 + exp((real(x, 8) / -real(s, 8))))
    code = real(tmp, 4)
end function

Julia

function code(x, s)
	tmp = Float64(1.0 / Float64(1.0 + exp(Float64(Float64(x) / Float64(-Float64(s))))))
	return Float32(tmp)
end

MATLAB

function tmp_2 = code(x, s)
	tmp = 1.0 / (1.0 + exp((double(x) / -s)));
	tmp_2 = single(tmp);
end

Derivation

1. Initial program 100.0%

   \[\langle \frac{1}{1 + e^{\frac{x}{-s}}} \rangle_{\text{binary64}} \]

2. Final simplification 100.0%

   \[\leadsto \langle \frac{1}{1 + e^{\frac{x}{-s}}} \rangle_{\text{binary64}} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. inv-pow 99.8%

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

   2. metadata-eval 99.8%

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

   3. pow-to-exp 99.9%

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

   4. exp-prod 99.8%

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

   5. sqr-pow 99.3%

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

   6. log1p-def 99.3%

      \[\leadsto {\left(e^{\color{blue}{\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(e^{\log \left(1 + e^{\frac{-x}{s}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \]

   7. remove-double-neg 99.3%

      \[\leadsto {\left(e^{\mathsf{log1p}\left(e^{\frac{-x}{\color{blue}{-\left(-s\right)}}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(e^{\log \left(1 + e^{\frac{-x}{s}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \]

   8. frac-2neg 99.3%

      \[\leadsto {\left(e^{\mathsf{log1p}\left(e^{\color{blue}{\frac{x}{-s}}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(e^{\log \left(1 + e^{\frac{-x}{s}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \]

   9. metadata-eval 99.3%

      \[\leadsto {\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \cdot {\left(e^{\log \left(1 + e^{\frac{-x}{s}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \]

   10. metadata-eval 99.3%

       \[\leadsto {\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{\color{blue}{-0.5}} \cdot {\left(e^{\log \left(1 + e^{\frac{-x}{s}}\right)}\right)}^{\left(\frac{-1}{2}\right)} \]

3. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{-0.5} \cdot {\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{-0.5}} \]
4. Step-by-step derivation

   1. pow-sqr 99.8%

      \[\leadsto \color{blue}{{\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{\left(2 \cdot -0.5\right)}} \]

   2. metadata-eval 99.8%

      \[\leadsto {\left(e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}\right)}^{\color{blue}{-1}} \]

   3. unpow-1 99.8%

      \[\leadsto \color{blue}{\frac{1}{e^{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}}} \]

   4. exp-neg 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 4: 65.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.7%

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

      1. flip3-+ 42.5%

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

      2. metadata-eval 42.5%

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

      3. pow-exp 42.4%

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

      4. neg-mul-1 42.4%

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

      5. metadata-eval 42.4%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.4%

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

      8. metadata-eval 42.4%

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

      9. metadata-eval 42.4%

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

      10. metadata-eval 42.4%

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

      11. +-commutative 42.4%

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

   3. Applied egg-rr 42.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in x around 0 80.0%

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

      1. associate-+r+ 80.0%

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

      2. mul-1-neg 80.0%

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

      3. sub-neg 80.0%

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

      4. *-commutative 80.0%

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

      5. unpow2 80.0%

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

      6. unpow2 80.0%

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

   6. Simplified 80.0%

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

      1. clear-num 80.0%

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

      2. rgt-mult-inverse 80.0%

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

      3. frac-2neg 80.0%

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

      4. remove-double-neg 80.0%

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

      5. clear-num 80.0%

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

      6. times-frac 82.5%

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

      7. frac-times 82.5%

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

      8. associate-/r* 83.8%

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

      9. associate-/r* 83.8%

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

      10. associate-*l/ 83.8%

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

      11. frac-2neg 83.8%

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

      12. distribute-lft-neg-out 83.8%

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

      13. remove-double-neg 83.8%

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

      14. neg-mul-1 83.8%

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

      15. times-frac 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. times-frac 89.7%

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

      2. neg-mul-1 89.7%

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

      3. associate-/l/ 89.7%

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

      4. *-inverses 89.7%

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

      5. metadata-eval 89.7%

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

      6. associate-/r/ 89.7%

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

      7. associate-*l/ 89.7%

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

      8. associate-*r/ 89.7%

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

      9. metadata-eval 89.7%

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

      10. *-inverses 89.7%

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

      11. associate-/l/ 89.7%

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

      12. neg-mul-1 89.7%

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

      13. *-commutative 89.7%

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

      14. /-rgt-identity 89.7%

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

      15. associate-/l* 89.7%

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

      16. neg-mul-1 89.7%

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

      17. associate-/l/ 89.7%

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

      18. *-inverses 89.7%

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

      19. metadata-eval 89.7%

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

      20. metadata-eval 89.7%

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

      21. associate-/l/ 89.7%

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

      22. neg-mul-1 89.7%

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

   10. Simplified 89.7%

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

4. Final simplification 63.0%

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

Alternative 5: 62.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) - 0.5 \cdot \left(\frac{-1}{s \cdot s} \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999998199587e-24)
   0.5
   (/ 1.0 (- (- 2.0 (/ x s)) (* 0.5 (* (/ -1.0 (* s s)) (* x x)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((2.0f - (x / s)) - (0.5f * ((-1.0f / (s * s)) * (x * x))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((2.0e0 - (x / s)) - (0.5e0 * (((-1.0e0) / (s * s)) * (x * x))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(x / s)) - Float32(Float32(0.5) * Float32(Float32(Float32(-1.0) / Float32(s * s)) * Float32(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(2.0) - (x / s)) - (single(0.5) * ((single(-1.0) / (s * s)) * (x * x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1e-23

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.4%

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

   if 1e-23 < (neg.f32 x)

   1. Initial program 99.9%

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

      1. flip3-+ 10.0%

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

      2. metadata-eval 10.0%

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

      3. pow-exp 9.9%

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

      4. neg-mul-1 9.9%

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

      5. metadata-eval 9.9%

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

      6. associate-/l* 9.9%

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

      7. associate-*l/ 9.9%

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

      8. metadata-eval 9.9%

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

      9. metadata-eval 9.9%

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

      10. metadata-eval 9.9%

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

      11. +-commutative 9.9%

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

   3. Applied egg-rr 10.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in x around 0 83.8%

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

      1. associate-+r+ 83.8%

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

      2. mul-1-neg 83.8%

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

      3. sub-neg 83.8%

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

      4. *-commutative 83.8%

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

      5. unpow2 83.8%

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

      6. unpow2 83.8%

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

   6. Simplified 83.8%

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

      1. frac-2neg 83.8%

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

      2. div-inv 83.8%

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

      3. distribute-rgt-neg-in 83.8%

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

      4. frac-2neg 83.8%

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

      5. metadata-eval 83.8%

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

      6. distribute-rgt-neg-in 83.8%

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

      7. distribute-rgt-neg-in 83.8%

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

      8. remove-double-neg 83.8%

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

   8. Applied egg-rr 83.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) - 0.5 \cdot \left(\frac{-1}{s \cdot s} \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]

Alternative 6: 61.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999998199587e-24)
   0.5
   (/ 1.0 (+ (- 2.0 (/ x s)) (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((2.0f - (x / s)) + (0.5f * ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((2.0e0 - (x / s)) + (0.5e0 * ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(2.0) - Float32(x / s)) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(2.0) - (x / s)) + (single(0.5) * ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1e-23

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.4%

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

   if 1e-23 < (neg.f32 x)

   1. Initial program 99.9%

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

      1. flip3-+ 10.0%

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

      2. metadata-eval 10.0%

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

      3. pow-exp 9.9%

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

      4. neg-mul-1 9.9%

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

      5. metadata-eval 9.9%

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

      6. associate-/l* 9.9%

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

      7. associate-*l/ 9.9%

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

      8. metadata-eval 9.9%

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

      9. metadata-eval 9.9%

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

      10. metadata-eval 9.9%

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

      11. +-commutative 9.9%

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

   3. Applied egg-rr 10.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in x around 0 83.8%

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

      1. associate-+r+ 83.8%

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

      2. mul-1-neg 83.8%

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

      3. sub-neg 83.8%

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

      4. *-commutative 83.8%

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

      5. unpow2 83.8%

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

      6. unpow2 83.8%

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

   6. Simplified 83.8%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(2 - \frac{x}{s}\right) + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 7: 61.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999998199587e-24)
   0.5
   (/ 1.0 (+ 2.0 (* 0.5 (/ (* x x) (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + (0.5f * ((x * x) / (s * s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * ((x * x) / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(x * x) / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + (single(0.5) * ((x * x) / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1e-23

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 47.4%

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

   if 1e-23 < (neg.f32 x)

   1. Initial program 99.9%

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

      1. flip3-+ 10.0%

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

      2. metadata-eval 10.0%

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

      3. pow-exp 9.9%

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

      4. neg-mul-1 9.9%

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

      5. metadata-eval 9.9%

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

      6. associate-/l* 9.9%

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

      7. associate-*l/ 9.9%

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

      8. metadata-eval 9.9%

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

      9. metadata-eval 9.9%

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

      10. metadata-eval 9.9%

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

      11. +-commutative 9.9%

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

   3. Applied egg-rr 10.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in x around 0 83.8%

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

      1. associate-+r+ 83.8%

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

      2. mul-1-neg 83.8%

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

      3. sub-neg 83.8%

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

      4. *-commutative 83.8%

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

      5. unpow2 83.8%

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

      6. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in x around 0 82.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{s \cdot s}}\\ \end{array} \]

Alternative 8: 49.2% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.7%

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

      1. flip3-+ 42.5%

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

      2. metadata-eval 42.5%

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

      3. pow-exp 42.4%

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

      4. neg-mul-1 42.4%

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

      5. metadata-eval 42.4%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.4%

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

      8. metadata-eval 42.4%

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

      9. metadata-eval 42.4%

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

      10. metadata-eval 42.4%

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

      11. +-commutative 42.4%

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

   3. Applied egg-rr 42.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in s around inf 43.0%

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

      1. associate--l+ 43.0%

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

      2. associate-*r/ 43.0%

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

      3. *-commutative 43.0%

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

      4. associate-*r/ 43.0%

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

      5. associate-*r/ 43.0%

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

      6. distribute-rgt-out-- 43.0%

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

      7. metadata-eval 43.0%

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

      8. *-commutative 43.0%

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

      9. neg-mul-1 43.0%

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

      10. associate-*l/ 43.0%

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

      11. neg-mul-1 43.0%

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

      12. associate-*l* 43.0%

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

      13. associate-*l/ 43.0%

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

      14. metadata-eval 43.0%

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

      15. *-commutative 43.0%

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

   6. Simplified 43.0%

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

      1. distribute-lft-out-- 61.6%

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

      2. *-commutative 61.6%

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

      3. sub-div 61.6%

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-3 - -2}{s}} \cdot x} \]

      4. metadata-eval 61.6%

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

   8. Applied egg-rr 61.6%

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

4. Final simplification 47.1%

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

Alternative 9: 49.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.7%

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

      1. flip3-+ 42.5%

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

      2. metadata-eval 42.5%

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

      3. pow-exp 42.4%

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

      4. neg-mul-1 42.4%

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

      5. metadata-eval 42.4%

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

      6. associate-/l* 42.4%

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

      7. associate-*l/ 42.4%

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

      8. metadata-eval 42.4%

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

      9. metadata-eval 42.4%

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

      10. metadata-eval 42.4%

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

      11. +-commutative 42.4%

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

   3. Applied egg-rr 42.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in s around inf 43.0%

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

      1. associate--l+ 43.0%

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

      2. associate-*r/ 43.0%

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

      3. *-commutative 43.0%

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

      4. associate-*r/ 43.0%

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

      5. associate-*r/ 43.0%

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

      6. distribute-rgt-out-- 43.0%

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

      7. metadata-eval 43.0%

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

      8. *-commutative 43.0%

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

      9. neg-mul-1 43.0%

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

      10. associate-*l/ 43.0%

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

      11. neg-mul-1 43.0%

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

      12. associate-*l* 43.0%

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

      13. associate-*l/ 43.0%

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

      14. metadata-eval 43.0%

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

      15. *-commutative 43.0%

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

   6. Simplified 43.0%

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

      1. distribute-lft-out-- 61.6%

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

      2. sub-div 61.6%

         \[\leadsto \frac{1}{2 + x \cdot \color{blue}{\frac{-3 - -2}{s}}} \]

      3. metadata-eval 61.6%

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

      4. metadata-eval 61.6%

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

      5. remove-double-neg 61.6%

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

      6. frac-2neg 61.6%

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

      7. div-inv 61.6%

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

      8. clear-num 61.6%

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

      9. metadata-eval 61.6%

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

      10. metadata-eval 61.6%

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

      11. distribute-frac-neg 61.6%

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

      12. frac-2neg 61.6%

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

      13. metadata-eval 61.6%

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

   8. Applied egg-rr 61.6%

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

4. Final simplification 47.1%

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

Alternative 10: 60.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 4.0) 0.5 (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 4.0f) {
		tmp = 0.5f;
	} else {
		tmp = (2.0f * (s * s)) / (x * x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 4.0e0) then
        tmp = 0.5e0
    else
        tmp = (2.0e0 * (s * s)) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(s * s)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(4.0))
		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) < 4

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 49.3%

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

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

   1. Initial program 100.0%

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

      1. flip3-+ -0.0%

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

      2. metadata-eval -0.0%

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

      3. pow-exp -0.0%

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

      4. neg-mul-1 -0.0%

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

      5. metadata-eval -0.0%

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

      6. associate-/l* -0.0%

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

      7. associate-*l/ -0.0%

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

      8. metadata-eval -0.0%

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

      9. metadata-eval -0.0%

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

      10. metadata-eval -0.0%

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

      11. +-commutative -0.0%

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

   3. Applied egg-rr -0.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + e^{\frac{-3}{\frac{s}{x}}}}{e^{\frac{-2}{\frac{s}{x}}} - \mathsf{expm1}\left(\frac{x}{-s}\right)}}} \]

   4. Taylor expanded in x around 0 81.3%

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

      1. associate-+r+ 81.3%

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

      2. mul-1-neg 81.3%

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

      3. sub-neg 81.3%

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

      4. *-commutative 81.3%

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

      5. unpow2 81.3%

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

      6. unpow2 81.3%

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

   6. Simplified 81.3%

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

   7. Taylor expanded in x around inf 79.7%

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

      1. unpow2 79.7%

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

      2. unpow2 79.7%

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

      3. associate-*r/ 79.7%

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

   9. Simplified 79.7%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]

Alternative 11: 49.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-2.0))
		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) < -2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 61.6%

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

      1. mul-1-neg 61.6%

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

      2. unsub-neg 61.6%

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

   4. Simplified 61.6%

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

4. Final simplification 47.1%

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

Alternative 12: 47.6% accurate, 10.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 49.4%

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

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 39.8%

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

      1. mul-1-neg 39.8%

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

      2. unsub-neg 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in x around inf 39.8%

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

      1. associate-*r/ 39.8%

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

      2. neg-mul-1 39.8%

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

   7. Simplified 39.8%

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

      1. associate-/r/ 34.7%

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

      2. frac-2neg 34.7%

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

      3. metadata-eval 34.7%

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

      4. remove-double-neg 34.7%

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

   9. Applied egg-rr 34.7%

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

       1. associate-*l/ 34.7%

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

       2. associate-/l* 39.8%

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

   11. Applied egg-rr 39.8%

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

4. Final simplification 46.2%

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

Alternative 13: 46.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.999999987845058e-8) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.999999987845058e-8f) {
		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 <= (-1.999999987845058e-8)) 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(-1.999999987845058e-8))
		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(-1.999999987845058e-8))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-8

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 45.6%

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

      1. mul-1-neg 45.6%

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

      2. unsub-neg 45.6%

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

   4. Simplified 45.6%

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

   5. Taylor expanded in x around inf 39.6%

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

      1. associate-*r/ 39.6%

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

      2. neg-mul-1 39.6%

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

   7. Simplified 39.6%

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

   if -1.99999999e-8 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 14: 35.0% 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 99.8%

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

2. Taylor expanded in x around 0 35.3%

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

3. Final simplification 35.3%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))