Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.9s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 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 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

6. Applied egg-rr 99.9%

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

   1. exp-1-e 99.9%

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

8. Simplified 99.9%

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

Alternative 2: 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 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. exp-prod 99.9%

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

6. Applied egg-rr 99.9%

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

   1. exp-1-e 99.9%

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

8. Simplified 99.9%

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

   1. pow-flip 99.9%

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

   2. distribute-neg-frac2 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (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. Step-by-step derivation

   1. distribute-frac-neg 99.8%

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

   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. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(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 5: 85.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;t\_0 \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -1.0)
     (+ 1.0 (/ s x))
     (if (<= t_0 4.999999840142846e+37)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 1.0f + (s / x);
	} else if (t_0 <= 4.999999840142846e+37f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = x / -s
    if (t_0 <= (-1.0e0)) then
        tmp = 1.0e0 + (s / x)
    else if (t_0 <= 4.999999840142846e+37) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (t_0 <= Float32(4.999999840142846e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(1.0) + (s / x);
	elseif (t_0 <= single(4.999999840142846e+37))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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 95.0%

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

   6. Taylor expanded in x around inf 94.9%

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

      1. mul-1-neg 94.9%

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

      2. unsub-neg 94.9%

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

   8. Simplified 94.9%

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

      1. sub-neg 94.9%

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

      2. distribute-frac-neg 94.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 96.1%

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

      5. sqr-neg 96.1%

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

      6. sqrt-unprod 95.2%

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

      7. add-sqr-sqrt 95.2%

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

      8. +-commutative 95.2%

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

   10. Applied egg-rr 95.2%

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

   if -1 < (/.f32 (neg.f32 x) s) < 4.99999984e37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.9%

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

      1. mul-1-neg 45.9%

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

      2. unsub-neg 45.9%

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

   5. Simplified 45.9%

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

      1. *-un-lft-identity 45.9%

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

      2. cancel-sign-sub-inv 45.9%

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

      3. metadata-eval 45.9%

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

      4. add-log-exp 94.5%

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

      5. pow-exp 94.5%

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

      6. flip-+ 39.6%

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

      7. metadata-eval 39.6%

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

      8. pow-exp 39.6%

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

      9. add-log-exp 39.6%

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

      10. neg-mul-1 39.6%

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

      11. pow-exp 39.6%

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

      12. add-log-exp 40.1%

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

      13. neg-mul-1 40.1%

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

      14. distribute-neg-frac 40.1%

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

      15. distribute-neg-frac 40.1%

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

      16. pow-exp 40.1%

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

   7. Applied egg-rr 79.9%

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

   if 4.99999984e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -1:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;\frac{x}{-s} \leq 4.999999840142846 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000544904023 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x \cdot 0.5 - s}{s \cdot \left(x \cdot -0.5\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000544904023e-27)
   (/ 1.0 (* x (/ (- (* x 0.5) s) (* s (* x -0.5)))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000544904023e-27f) {
		tmp = 1.0f / (x * (((x * 0.5f) - s) / (s * (x * -0.5f))));
	} 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 <= (-2.0000000544904023e-27)) then
        tmp = 1.0e0 / (x * (((x * 0.5e0) - s) / (s * (x * (-0.5e0)))))
    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(-2.0000000544904023e-27))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(x * Float32(0.5)) - s) / Float32(s * Float32(x * Float32(-0.5))))));
	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(-2.0000000544904023e-27))
		tmp = single(1.0) / (x * (((x * single(0.5)) - s) / (s * (x * single(-0.5)))));
	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 < -2.00000005e-27

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 37.6%

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

      1. mul-1-neg 37.6%

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

      2. unsub-neg 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in x around inf 37.5%

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

      1. sub-neg 37.5%

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

      2. associate-*r/ 37.5%

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

      3. metadata-eval 37.5%

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

      4. distribute-neg-frac 37.5%

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

      5. metadata-eval 37.5%

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

   8. Simplified 37.5%

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

      1. clear-num 37.5%

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

      2. clear-num 37.5%

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

      3. frac-add 41.9%

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

      4. *-un-lft-identity 41.9%

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

      5. div-inv 41.9%

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

      6. metadata-eval 41.9%

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

      7. div-inv 41.9%

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

      8. metadata-eval 41.9%

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

      9. div-inv 41.9%

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

      10. metadata-eval 41.9%

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

      11. div-inv 41.9%

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

      12. metadata-eval 41.9%

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

   10. Applied egg-rr 41.9%

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

       1. *-rgt-identity 41.9%

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

       2. +-commutative 41.9%

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

       3. *-commutative 41.9%

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

       4. mul-1-neg 41.9%

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

       5. unsub-neg 41.9%

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

       6. *-commutative 41.9%

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

       7. associate-*l* 41.9%

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

       8. *-commutative 41.9%

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

       9. associate-*l* 41.9%

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

       10. metadata-eval 41.9%

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

   12. Simplified 41.9%

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

   if -2.00000005e-27 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 93.2%

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

Alternative 7: 70.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{-1}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -9.999999682655225e-20)
   (+ 1.0 (/ s x))
   (if (<= (- x) 9.999999974752427e-7)
     (+ 0.5 (* (/ x s) 0.25))
     (/ 1.0 (* x (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -9.999999682655225e-20f) {
		tmp = 1.0f + (s / x);
	} else if (-x <= 9.999999974752427e-7f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 1.0f / (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 <= (-9.999999682655225e-20)) then
        tmp = 1.0e0 + (s / x)
    else if (-x <= 9.999999974752427e-7) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 1.0e0 / (x * ((-1.0e0) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-9.999999682655225e-20))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (Float32(-x) <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(1.0) / 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(-9.999999682655225e-20))
		tmp = single(1.0) + (s / x);
	elseif (-x <= single(9.999999974752427e-7))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / (x * (single(-1.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -9.99999968e-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 95.9%

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

   6. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   8. Simplified 86.9%

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

      1. sub-neg 86.9%

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

      2. distribute-frac-neg 86.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 88.4%

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

      5. sqr-neg 88.4%

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

      6. sqrt-unprod 88.4%

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

      7. add-sqr-sqrt 88.4%

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

      8. +-commutative 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -9.99999968e-20 < (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 62.0%

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

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in x around inf 38.3%

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

      1. sub-neg 38.3%

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

      2. associate-*r/ 38.3%

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

      3. metadata-eval 38.3%

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

      4. distribute-neg-frac 38.3%

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

      5. metadata-eval 38.3%

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

   8. Simplified 38.3%

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

   9. Taylor expanded in x around inf 38.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{-1}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -9.999999682655225e-20)
   (+ 1.0 (/ s x))
   (if (<= (- x) 9.999999974752427e-7)
     (+ 0.5 (* (/ x s) 0.25))
     (/ -1.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -9.999999682655225e-20f) {
		tmp = 1.0f + (s / x);
	} else if (-x <= 9.999999974752427e-7f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} 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 <= (-9.999999682655225e-20)) then
        tmp = 1.0e0 + (s / x)
    else if (-x <= 9.999999974752427e-7) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    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(-9.999999682655225e-20))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (Float32(-x) <= Float32(9.999999974752427e-7))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	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(-9.999999682655225e-20))
		tmp = single(1.0) + (s / x);
	elseif (-x <= single(9.999999974752427e-7))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -9.99999968e-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 95.9%

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

   6. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   8. Simplified 86.9%

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

      1. sub-neg 86.9%

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

      2. distribute-frac-neg 86.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 88.4%

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

      5. sqr-neg 88.4%

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

      6. sqrt-unprod 88.4%

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

      7. add-sqr-sqrt 88.4%

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

      8. +-commutative 88.4%

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

   10. Applied egg-rr 88.4%

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

   if -9.99999968e-20 < (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 62.0%

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

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in x around inf 38.3%

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

      1. mul-1-neg 38.3%

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

      2. distribute-neg-frac2 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000544904023 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot 2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000544904023e-27)
   (/ 1.0 (* x (/ (+ x (* s 2.0)) (* x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000544904023e-27f) {
		tmp = 1.0f / (x * ((x + (s * 2.0f)) / (x * s)));
	} 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 <= (-2.0000000544904023e-27)) then
        tmp = 1.0e0 / (x * ((x + (s * 2.0e0)) / (x * s)))
    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(-2.0000000544904023e-27))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x + Float32(s * Float32(2.0))) / Float32(x * s))));
	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(-2.0000000544904023e-27))
		tmp = single(1.0) / (x * ((x + (s * single(2.0))) / (x * s)));
	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 < -2.00000005e-27

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 37.6%

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

      1. mul-1-neg 37.6%

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

      2. unsub-neg 37.6%

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

   5. Simplified 37.6%

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

   6. Taylor expanded in x around inf 37.5%

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

      1. sub-neg 37.5%

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

      2. associate-*r/ 37.5%

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

      3. metadata-eval 37.5%

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

      4. distribute-neg-frac 37.5%

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

      5. metadata-eval 37.5%

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

   8. Simplified 37.5%

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

      1. +-commutative 37.5%

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

      2. frac-2neg 37.5%

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

      3. metadata-eval 37.5%

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

      4. frac-add 42.1%

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

      5. *-un-lft-identity 42.1%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 35.0%

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

      8. sqr-neg 35.0%

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

      9. sqrt-unprod 35.0%

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

      10. add-sqr-sqrt 35.0%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 59.3%

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

      13. sqr-neg 59.3%

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

      14. sqrt-unprod 41.1%

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

      15. add-sqr-sqrt 41.2%

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

   10. Applied egg-rr 41.2%

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

   if -2.00000005e-27 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 93.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000544904023 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot 2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -2.99999994735501 \cdot 10^{-14}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -2.99999994735501e-14)
   (+ 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 <= -2.99999994735501e-14f) {
		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 <= (-2.99999994735501e-14)) 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(-2.99999994735501e-14))
		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(-2.99999994735501e-14))
		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) < -2.99999995e-14

   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 97.7%

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

   6. Taylor expanded in x around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. unsub-neg 92.2%

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

   8. Simplified 92.2%

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

      1. sub-neg 92.2%

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

      2. distribute-frac-neg 92.2%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 93.2%

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

      5. sqr-neg 93.2%

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

      6. sqrt-unprod 93.2%

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

      7. add-sqr-sqrt 93.2%

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

      8. +-commutative 93.2%

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

   10. Applied egg-rr 93.2%

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

   if -2.99999995e-14 < (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 57.6%

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

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in x around inf 38.3%

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

      1. mul-1-neg 38.3%

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

      2. distribute-neg-frac2 38.3%

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

   8. Simplified 38.3%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -2.99999994735501 \cdot 10^{-14}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -2.99999994735501 \cdot 10^{-14}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -2.99999994735501e-14)
   (+ 1.0 (/ s x))
   (if (<= (- x) 9.999999974752427e-7) 0.5 (/ s (- x)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -2.99999994735501e-14f) {
		tmp = 1.0f + (s / x);
	} else if (-x <= 9.999999974752427e-7f) {
		tmp = 0.5f;
	} else {
		tmp = 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 <= (-2.99999994735501e-14)) then
        tmp = 1.0e0 + (s / x)
    else if (-x <= 9.999999974752427e-7) then
        tmp = 0.5e0
    else
        tmp = s / -x
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-2.99999994735501e-14))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (Float32(-x) <= Float32(9.999999974752427e-7))
		tmp = Float32(0.5);
	else
		tmp = Float32(s / Float32(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-2.99999994735501e-14))
		tmp = single(1.0) + (s / x);
	elseif (-x <= single(9.999999974752427e-7))
		tmp = single(0.5);
	else
		tmp = s / -x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (neg.f32 x) < -2.99999995e-14

   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 97.7%

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

   6. Taylor expanded in x around inf 92.2%

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

      1. mul-1-neg 92.2%

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

      2. unsub-neg 92.2%

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

   8. Simplified 92.2%

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

      1. sub-neg 92.2%

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

      2. distribute-frac-neg 92.2%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 93.2%

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

      5. sqr-neg 93.2%

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

      6. sqrt-unprod 93.2%

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

      7. add-sqr-sqrt 93.2%

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

      8. +-commutative 93.2%

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

   10. Applied egg-rr 93.2%

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

   if -2.99999995e-14 < (neg.f32 x) < 9.99999997e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 57.6%

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

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in x around inf 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

   8. Simplified 35.3%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -2.99999994735501 \cdot 10^{-14}:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;-x \leq 9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6000000081769018 \cdot 10^{-37}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.6000000081769018e-37)
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.6000000081769018e-37f) {
		tmp = 1.0f / (-1.0f + (3.0f - (x / s)));
	} 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 <= (-1.6000000081769018e-37)) then
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / s)))
    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(-1.6000000081769018e-37))
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / s))));
	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(-1.6000000081769018e-37))
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	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 < -1.60000001e-37

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 41.0%

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

      1. mul-1-neg 41.0%

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

      2. unsub-neg 41.0%

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

   5. Simplified 41.0%

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

      1. expm1-log1p-u 41.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]

      2. expm1-undefine 41.0%

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

   7. Applied egg-rr 41.0%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} - 1}} \]
   8. Step-by-step derivation

      1. sub-neg 41.0%

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

      2. metadata-eval 41.0%

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

      3. +-commutative 41.0%

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

      4. log1p-undefine 41.0%

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

      5. rem-exp-log 41.0%

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

      6. associate-+r- 41.0%

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

      7. metadata-eval 41.0%

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

   9. Simplified 41.0%

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

   if -1.60000001e-37 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      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 94.3%

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

Alternative 13: 70.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.9999998413276127e-20)
   (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))
   (+ 1.0 (/ s x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.9999998413276127e-20f) {
		tmp = 1.0f / (-1.0f + (3.0f - (x / 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 <= 4.9999998413276127e-20) then
        tmp = 1.0e0 / ((-1.0e0) + (3.0e0 - (x / 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(4.9999998413276127e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(-1.0) + Float32(Float32(3.0) - Float32(x / 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(4.9999998413276127e-20))
		tmp = single(1.0) / (single(-1.0) + (single(3.0) - (x / s)));
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999984e-20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. unsub-neg 49.0%

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

   5. Simplified 49.0%

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

      1. expm1-log1p-u 48.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]

      2. expm1-undefine 48.8%

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

   7. Applied egg-rr 48.8%

      \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} - 1}} \]
   8. Step-by-step derivation

      1. sub-neg 48.8%

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

      2. metadata-eval 48.8%

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

      3. +-commutative 48.8%

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

      4. log1p-undefine 48.8%

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

      5. rem-exp-log 48.9%

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

      6. associate-+r- 49.0%

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

      7. metadata-eval 49.0%

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

   9. Simplified 49.0%

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

   if 4.99999984e-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 95.9%

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

   6. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   8. Simplified 86.9%

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

      1. sub-neg 86.9%

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

      2. distribute-frac-neg 86.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 88.4%

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

      5. sqr-neg 88.4%

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

      6. sqrt-unprod 88.4%

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

      7. add-sqr-sqrt 88.4%

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

      8. +-commutative 88.4%

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

   10. Applied egg-rr 88.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.9999998413276127e-20) (/ 1.0 (- 2.0 (/ x s))) (+ 1.0 (/ s x))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.9999998413276127e-20f) {
		tmp = 1.0f / (2.0f - (x / 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 <= 4.9999998413276127e-20) then
        tmp = 1.0e0 / (2.0e0 - (x / 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(4.9999998413276127e-20))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / 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(4.9999998413276127e-20))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999984e-20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 49.0%

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

      1. mul-1-neg 49.0%

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

      2. unsub-neg 49.0%

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

   5. Simplified 49.0%

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

   if 4.99999984e-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 95.9%

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

   6. Taylor expanded in x around inf 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   8. Simplified 86.9%

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

      1. sub-neg 86.9%

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

      2. distribute-frac-neg 86.9%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 88.4%

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

      5. sqr-neg 88.4%

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

      6. sqrt-unprod 88.4%

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

      7. add-sqr-sqrt 88.4%

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

      8. +-commutative 88.4%

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

   10. Applied egg-rr 88.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.9999998413276127 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.4% accurate, 10.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-x <= 9.999999974752427e-7f) {
		tmp = 0.5f;
	} else {
		tmp = 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 = 0.5e0
    else
        tmp = 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(0.5);
	else
		tmp = Float32(s / Float32(-x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(9.999999974752427e-7))
		tmp = single(0.5);
	else
		tmp = s / -x;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 43.9%

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

   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 38.3%

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

      1. mul-1-neg 38.3%

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

      2. unsub-neg 38.3%

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

   5. Simplified 38.3%

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

   6. Taylor expanded in x around inf 35.3%

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

      1. mul-1-neg 35.3%

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

      2. distribute-neg-frac2 35.3%

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

   8. Simplified 35.3%

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

Alternative 16: 34.7% 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 32.5%

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

Reproduce

herbie shell --seed 2024095 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))