Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 9.6s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. div-inv 99.8%

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

   2. exp-prod 81.4%

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

   3. neg-mul-1 81.4%

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

   4. exp-prod 81.4%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2000000136174153 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.2000000136174153e-17)
   (/ (/ (pow s 2.0) (- s)) x)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.2000000136174153e-17f) {
		tmp = (powf(s, 2.0f) / -s) / x;
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (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 <= (-1.2000000136174153e-17)) then
        tmp = ((s ** 2.0e0) / -s) / x
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.2000000136174153e-17))
		tmp = Float32(Float32((s ^ Float32(2.0)) / Float32(-s)) / x);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.2000000136174153e-17))
		tmp = ((s ^ single(2.0)) / -s) / x;
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.20000001e-17

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 40.5%

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

      1. neg-mul-1 40.5%

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

      2. unsub-neg 40.5%

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

   5. Simplified 40.5%

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

   6. Taylor expanded in x around inf 30.7%

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

      1. associate-*r/ 30.7%

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

      2. neg-mul-1 30.7%

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

   8. Simplified 30.7%

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

      1. neg-sub0 30.7%

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

      2. flip-- 54.1%

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

      3. metadata-eval 54.1%

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

      4. pow2 54.1%

         \[\leadsto \frac{\frac{0 - \color{blue}{{s}^{2}}}{0 + s}}{x} \]

      5. add-sqr-sqrt 54.1%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      6. sqrt-unprod 11.2%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{s \cdot s}}}}{x} \]

      7. sqr-neg 11.2%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \sqrt{\color{blue}{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      8. sqrt-unprod -0.0%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      9. add-sqr-sqrt 52.5%

         \[\leadsto \frac{\frac{0 - {s}^{2}}{0 + \color{blue}{\left(-s\right)}}}{x} \]

      10. sub-neg 52.5%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{0 - s}}}{x} \]

      11. neg-sub0 52.5%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{-s}}}{x} \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{x} \]

      13. sqrt-unprod 11.2%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{x} \]

      14. sqr-neg 11.2%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\sqrt{\color{blue}{s \cdot s}}}}{x} \]

      15. sqrt-unprod 54.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{x} \]

      16. add-sqr-sqrt 54.1%

          \[\leadsto \frac{\frac{0 - {s}^{2}}{\color{blue}{s}}}{x} \]

   10. Applied egg-rr 54.1%

       \[\leadsto \frac{\color{blue}{\frac{0 - {s}^{2}}{s}}}{x} \]
   11. Step-by-step derivation

       1. sub0-neg 54.1%

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

   12. Simplified 54.1%

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

   if -1.20000001e-17 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 92.8%

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

      1. +-commutative 92.8%

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

   7. Simplified 92.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2000000136174153 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{{s}^{2}}{-s}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

6. Applied egg-rr 99.8%

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

   1. inv-pow 99.8%

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

   2. pow-pow 99.8%

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

   3. exp-1-e 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-commutative 99.8%

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

   2. neg-mul-1 99.8%

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

   3. distribute-neg-frac2 99.8%

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

10. Simplified 99.8%

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

Alternative 5: 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(x / Float32(-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. Add Preprocessing

3. Final simplification 99.8%

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

Alternative 6: 80.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 999999986991104:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -4.000000072010038e-35)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (if (<= (- x) 999999986991104.0)
     (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ (/ x s) 2.0)))
     (/ 1.0 (+ 2.0 (/ -1.0 (/ s x)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -4.000000072010038e-35f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (-x <= 999999986991104.0f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / ((x / s) + 2.0f));
	} 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 <= (-4.000000072010038e-35)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (-x <= 999999986991104.0e0) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / ((x / s) + 2.0e0))
    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(-x) <= Float32(-4.000000072010038e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (Float32(-x) <= Float32(999999986991104.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(x / s) + Float32(2.0))));
	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 <= single(-4.000000072010038e-35))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (-x <= single(999999986991104.0))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -4.00000007e-35

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   if -4.00000007e-35 < (neg.f32 x) < 9.99999987e14

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 47.2%

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

      1. neg-mul-1 47.2%

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

      2. unsub-neg 47.2%

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

   5. Simplified 47.2%

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

      1. *-un-lft-identity 47.2%

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

      2. cancel-sign-sub-inv 47.2%

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

      3. metadata-eval 47.2%

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

      4. add-log-exp 95.1%

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

      5. pow-exp 95.1%

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

      6. flip-+ 36.8%

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

      7. metadata-eval 36.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right) \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      8. pow-exp 36.8%

         \[\leadsto \frac{1}{\frac{4 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      9. add-log-exp 36.8%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-1 \cdot \frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      10. neg-mul-1 36.8%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      11. pow-exp 36.8%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      12. add-log-exp 37.7%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      13. neg-mul-1 37.7%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      14. distribute-neg-frac2 37.7%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      15. distribute-neg-frac2 37.7%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      16. pow-exp 37.7%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}} \]

   7. Applied egg-rr 64.1%

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

      1. frac-times 62.9%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x \cdot x}{\left(-s\right) \cdot \left(-s\right)}}}{2 - \frac{x}{-s}}} \]

      2. sqr-neg 62.9%

         \[\leadsto \frac{1}{\frac{4 - \frac{x \cdot x}{\color{blue}{s \cdot s}}}{2 - \frac{x}{-s}}} \]

      3. frac-times 64.1%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{s} \cdot \frac{x}{s}}}{2 - \frac{x}{-s}}} \]

      4. clear-num 64.1%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1}{\frac{s}{x}}} \cdot \frac{x}{s}}{2 - \frac{x}{-s}}} \]

      5. frac-times 66.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{1 \cdot x}{\frac{s}{x} \cdot s}}}{2 - \frac{x}{-s}}} \]

      6. *-un-lft-identity 66.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{x}}{\frac{s}{x} \cdot s}}{2 - \frac{x}{-s}}} \]

   9. Applied egg-rr 66.6%

      \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{\frac{s}{x} \cdot s}}}{2 - \frac{x}{-s}}} \]

   if 9.99999987e14 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.2%

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

      1. neg-mul-1 67.2%

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

      2. unsub-neg 67.2%

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

   5. Simplified 67.2%

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

      1. clear-num 67.2%

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

      2. inv-pow 67.2%

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

   7. Applied egg-rr 67.2%

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

      1. unpow-1 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 999999986991104:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 999999986991104:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 9.999999974752427e-7)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (if (<= (- x) 999999986991104.0)
     (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
     (/ 1.0 (+ 2.0 (/ -1.0 (/ s x)))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999974752427e-7f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (-x <= 999999986991104.0f) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} 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 <= 9.999999974752427e-7) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (-x <= 999999986991104.0e0) then
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (x / s))
    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(-x) <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (Float32(-x) <= Float32(999999986991104.0))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(x / s)));
	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 <= single(9.999999974752427e-7))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (-x <= single(999999986991104.0))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (x / s));
	else
		tmp = single(1.0) / (single(2.0) + (single(-1.0) / (s / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 82.0%

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

      1. +-commutative 82.0%

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

   7. Simplified 82.0%

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

   if 9.99999997e-7 < (neg.f32 x) < 9.99999987e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 23.5%

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

      1. neg-mul-1 23.5%

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

      2. unsub-neg 23.5%

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

   5. Simplified 23.5%

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

      1. *-un-lft-identity 23.5%

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

      2. cancel-sign-sub-inv 23.5%

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

      3. metadata-eval 23.5%

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

      4. add-log-exp 100.0%

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

      5. pow-exp 100.0%

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

      6. flip-+ -0.0%

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

      7. metadata-eval -0.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right) \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      8. pow-exp -0.0%

         \[\leadsto \frac{1}{\frac{4 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      9. add-log-exp -0.0%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-1 \cdot \frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      10. neg-mul-1 -0.0%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{s}\right)} \cdot \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      11. pow-exp -0.0%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      12. add-log-exp 0.9%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      13. neg-mul-1 0.9%

          \[\leadsto \frac{1}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      14. distribute-neg-frac2 0.9%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      15. distribute-neg-frac2 0.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \log \left({\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}\right)}} \]

      16. pow-exp 0.9%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \log \color{blue}{\left(e^{-1 \cdot \frac{x}{s}}\right)}}} \]

   7. Applied egg-rr 61.2%

      \[\leadsto \frac{1}{\color{blue}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}}} \]

   8. Taylor expanded in x around inf 61.2%

      \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{\color{blue}{\frac{x}{s}}}} \]

   if 9.99999987e14 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 67.2%

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

      1. neg-mul-1 67.2%

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

      2. unsub-neg 67.2%

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

   5. Simplified 67.2%

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

      1. clear-num 67.2%

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

      2. inv-pow 67.2%

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

   7. Applied egg-rr 67.2%

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

      1. unpow-1 67.2%

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

   9. Simplified 67.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;-x \leq 999999986991104:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{-1}{\frac{s}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1}{s}}\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6)
   (/ 1.0 (* x (/ 1.0 s)))
   (if (<= x 1.999999936531045e-20)
     (+ 0.5 (/ (* x 0.25) s))
     (/ 1.0 (+ 1.0 (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = 1.0f / (x * (1.0f / s));
	} else if (x <= 1.999999936531045e-20f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f / (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 <= (-1.9999999949504854e-6)) then
        tmp = 1.0e0 / (x * (1.0e0 / s))
    else if (x <= 1.999999936531045e-20) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 / (1.0e0 + (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(1.0) / s)));
	elseif (x <= Float32(1.999999936531045e-20))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = single(1.0) / (x * (single(1.0) / s));
	elseif (x <= single(1.999999936531045e-20))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / (single(1.0) + (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. distribute-frac-neg2 43.5%

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

   8. Simplified 43.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 59.0%

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

      3. sqr-neg 59.0%

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

      4. sqrt-unprod 43.5%

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

      5. add-sqr-sqrt 43.5%

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

      6. div-inv 43.5%

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

      7. *-commutative 43.5%

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

   10. Applied egg-rr 43.5%

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

   if -1.99999999e-6 < x < 1.99999994e-20

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.2%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 66.2%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   if 1.99999994e-20 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around inf 90.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1}{s}}\\ \mathbf{elif}\;x \leq 1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1}{s}}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6)
   (/ 1.0 (* x (/ 1.0 s)))
   (if (<= x 1.9999999920083944e-11)
     (+ 0.5 (/ (* x 0.25) s))
     (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = 1.0f / (x * (1.0f / s));
	} else if (x <= 1.9999999920083944e-11f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 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 <= (-1.9999999949504854e-6)) then
        tmp = 1.0e0 / (x * (1.0e0 / s))
    else if (x <= 1.9999999920083944e-11) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(1.0) / s)));
	elseif (x <= Float32(1.9999999920083944e-11))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = single(1.0) / (x * (single(1.0) / s));
	elseif (x <= single(1.9999999920083944e-11))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 43.5%

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

      1. mul-1-neg 43.5%

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

      2. distribute-frac-neg2 43.5%

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

   8. Simplified 43.5%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 59.0%

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

      3. sqr-neg 59.0%

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

      4. sqrt-unprod 43.5%

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

      5. add-sqr-sqrt 43.5%

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

      6. div-inv 43.5%

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

      7. *-commutative 43.5%

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

   10. Applied egg-rr 43.5%

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

   if -1.99999999e-6 < x < 1.99999999e-11

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 65.1%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.1%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   if 1.99999999e-11 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around inf 93.7%

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

      1. neg-mul-1 93.7%

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

      2. unsub-neg 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x \cdot \frac{1}{s}}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6)
   (/ 1.0 (/ x s))
   (if (<= x 1.9999999920083944e-11)
     (+ 0.5 (/ (* x 0.25) s))
     (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 1.9999999920083944e-11f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 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 <= (-1.9999999949504854e-6)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 1.9999999920083944e-11) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(1.9999999920083944e-11))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(1.9999999920083944e-11))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

      1. neg-sub0 39.4%

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

      2. sub-neg 39.4%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 57.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 57.2%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 39.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. +-lft-identity 39.4%

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

   12. Simplified 39.4%

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

       1. clear-num 43.5%

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

       2. inv-pow 43.5%

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

   14. Applied egg-rr 43.5%

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

       1. unpow-1 43.5%

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

   16. Simplified 43.5%

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

   if -1.99999999e-6 < x < 1.99999999e-11

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 65.1%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.1%

         \[\leadsto 0.5 + \color{blue}{\frac{0.25 \cdot x}{s}} \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{0.5 + \frac{0.25 \cdot x}{s}} \]

   if 1.99999999e-11 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around inf 93.7%

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

      1. neg-mul-1 93.7%

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

      2. unsub-neg 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -3.999999984016789 \cdot 10^{-11}:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -3.999999984016789e-11)
   (- 1.0 (/ s x))
   (if (<= (- x) 9.999999974752427e-7) 0.5 (/ 1.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -3.999999984016789e-11f) {
		tmp = 1.0f - (s / x);
	} else if (-x <= 9.999999974752427e-7f) {
		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 <= (-3.999999984016789e-11)) then
        tmp = 1.0e0 - (s / x)
    else if (-x <= 9.999999974752427e-7) 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(-x) <= Float32(-3.999999984016789e-11))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (Float32(-x) <= Float32(9.999999974752427e-7))
		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 <= single(-3.999999984016789e-11))
		tmp = single(1.0) - (s / x);
	elseif (-x <= single(9.999999974752427e-7))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -3.99999998e-11

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around inf 93.7%

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

      1. neg-mul-1 93.7%

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

      2. unsub-neg 93.7%

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

   10. Simplified 93.7%

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

   if -3.99999998e-11 < (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 59.5%

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

   if 9.99999997e-7 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

      1. neg-sub0 39.4%

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

      2. sub-neg 39.4%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 57.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 57.2%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 39.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. +-lft-identity 39.4%

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

   12. Simplified 39.4%

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

       1. clear-num 43.5%

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

       2. inv-pow 43.5%

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

   14. Applied egg-rr 43.5%

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

       1. unpow-1 43.5%

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

   16. Simplified 43.5%

       \[\leadsto \color{blue}{\frac{1}{\frac{x}{s}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{1}{s}}{\frac{-1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -4.000000072010038e-35)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (/ (/ 1.0 s) (/ -1.0 x))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -4.000000072010038e-35f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + ((1.0f / s) / (-1.0f / 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 <= (-4.000000072010038e-35)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + ((1.0e0 / s) / ((-1.0e0) / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-4.000000072010038e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(1.0) / s) / Float32(Float32(-1.0) / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-4.000000072010038e-35))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(1.0) / s) / (single(-1.0) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -4.00000007e-35

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   if -4.00000007e-35 < (neg.f32 x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 51.8%

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

      1. neg-mul-1 51.8%

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

      2. unsub-neg 51.8%

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

   5. Simplified 51.8%

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

      1. clear-num 51.8%

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

      2. associate-/r/ 51.8%

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

   7. Applied egg-rr 51.8%

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

      1. associate-/r/ 51.8%

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

      2. div-inv 51.8%

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

      3. associate-/r* 51.8%

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

   9. Applied egg-rr 51.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \frac{\frac{1}{s}}{\frac{-1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -4.000000072010038e-35)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -4.000000072010038e-35f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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 <= (-4.000000072010038e-35)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    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(-x) <= Float32(-4.000000072010038e-35))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	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 <= single(-4.000000072010038e-35))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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 (neg.f32 x) < -4.00000007e-35

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   if -4.00000007e-35 < (neg.f32 x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 51.8%

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

      1. neg-mul-1 51.8%

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

      2. unsub-neg 51.8%

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

   5. Simplified 51.8%

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

      1. clear-num 51.8%

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

      2. associate-/r/ 51.8%

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

   7. Applied egg-rr 51.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -4.000000072010038 \cdot 10^{-35}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6)
   (* s (/ 1.0 x))
   (if (<= x 1.9999999920083944e-11) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = s * (1.0f / x);
	} else if (x <= 1.9999999920083944e-11f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-1.9999999949504854e-6)) then
        tmp = s * (1.0e0 / x)
    else if (x <= 1.9999999920083944e-11) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	elseif (x <= Float32(1.9999999920083944e-11))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = s * (single(1.0) / x);
	elseif (x <= single(1.9999999920083944e-11))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

      1. clear-num 43.5%

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

      2. associate-/r/ 39.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 57.2%

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

      5. sqr-neg 57.2%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 39.4%

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

   10. Applied egg-rr 39.4%

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

   if -1.99999999e-6 < x < 1.99999999e-11

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 59.5%

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

   if 1.99999999e-11 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 96.7%

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

      1. +-commutative 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in x around inf 93.7%

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

      1. neg-mul-1 93.7%

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

      2. unsub-neg 93.7%

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

   10. Simplified 93.7%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.4% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.999999936531045e-20)
   (/ 1.0 (+ 1.0 (/ s x)))
   (/ 1.0 (+ 2.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.999999936531045e-20f) {
		tmp = 1.0f / (1.0f + (s / x));
	} 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 <= (-1.999999936531045e-20)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    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(-x) <= Float32(-1.999999936531045e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	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 <= single(-1.999999936531045e-20))
		tmp = single(1.0) / (single(1.0) + (s / x));
	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 (neg.f32 x) < -1.99999994e-20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around inf 90.4%

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

   if -1.99999994e-20 < (neg.f32 x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. unsub-neg 55.6%

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

   5. Simplified 55.6%

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

      1. clear-num 55.6%

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

      2. associate-/r/ 55.6%

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

   7. Applied egg-rr 55.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-1}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.999999936531045 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.999999936531045e-20)
   (/ 1.0 (+ 1.0 (/ s x)))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.999999936531045e-20f) {
		tmp = 1.0f / (1.0f + (s / x));
	} 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 <= (-1.999999936531045e-20)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    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(-x) <= Float32(-1.999999936531045e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	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 <= single(-1.999999936531045e-20))
		tmp = single(1.0) / (single(1.0) + (s / x));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1.99999994e-20

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 97.0%

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

      1. +-commutative 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around inf 90.4%

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

   if -1.99999994e-20 < (neg.f32 x)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. unsub-neg 55.6%

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

   5. Simplified 55.6%

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

Alternative 17: 45.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6) (* s (/ 1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = s * (1.0f / 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 <= (-1.9999999949504854e-6)) then
        tmp = s * (1.0e0 / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = s * (single(1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

      1. clear-num 43.5%

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

      2. associate-/r/ 39.4%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 57.2%

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

      5. sqr-neg 57.2%

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

      6. sqrt-unprod 39.4%

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

      7. add-sqr-sqrt 39.4%

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

   10. Applied egg-rr 39.4%

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

   if -1.99999999e-6 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 46.2%

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

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 45.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		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 <= (-1.9999999949504854e-6)) 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(-1.9999999949504854e-6))
		tmp = 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(-1.9999999949504854e-6))
		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-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 43.5%

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

      1. neg-mul-1 43.5%

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

      2. unsub-neg 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in x around inf 39.4%

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

      1. associate-*r/ 39.4%

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

      2. neg-mul-1 39.4%

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

   8. Simplified 39.4%

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

      1. neg-sub0 39.4%

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

      2. sub-neg 39.4%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 57.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 57.2%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 39.4%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 39.4%

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

   10. Applied egg-rr 39.4%

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

       1. +-lft-identity 39.4%

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

   12. Simplified 39.4%

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

   if -1.99999999e-6 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 46.2%

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

Alternative 19: 34.6% 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. Add Preprocessing

3. Taylor expanded in x around 0 35.0%

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

Reproduce

herbie shell --seed 2024166 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))