Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

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

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 60.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq 9.999999933815813 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -2000.0)
     (/ 1.0 (* x (/ 2.0 x)))
     (if (<= t_0 9.999999933815813e+36)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -2000.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (t_0 <= 9.999999933815813e+36f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (2.0f + (x / s)));
	} 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 <= (-2000.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (t_0 <= 9.999999933815813e+36) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (2.0e0 + (x / s)))
    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(-2000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(9.999999933815813e+36))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(2.0) + Float32(x / s))));
	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(-2000.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (t_0 <= single(9.999999933815813e+36))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (single(2.0) + (x / s)));
	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) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. neg-mul-1 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. sub-neg 5.1%

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

      2. associate-*r/ 5.1%

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

      3. metadata-eval 5.1%

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

      4. distribute-neg-frac 5.1%

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

      5. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -2e3 < (/.f32 (neg.f32 x) s) < 9.99999993e36

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 52.4%

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

      1. neg-mul-1 52.4%

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

      2. unsub-neg 52.4%

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

   5. Simplified 52.4%

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

      1. sub-neg 52.4%

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

      2. flip-+ 75.2%

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

      3. metadata-eval 75.2%

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

      4. distribute-neg-frac2 75.2%

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

      5. distribute-neg-frac2 75.2%

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

      6. distribute-neg-frac2 75.2%

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

   7. Applied egg-rr 75.2%

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

      1. distribute-frac-neg2 75.2%

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

      2. distribute-frac-neg2 75.2%

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

      3. sqr-neg 75.2%

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

      4. clear-num 75.2%

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

      5. frac-times 80.0%

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

      6. *-un-lft-identity 80.0%

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

   9. Applied egg-rr 80.0%

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

   if 9.99999993e36 < (/.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. neg-mul-1 100.0%

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

      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 93.1%

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

      1. associate-*r/ 93.1%

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

      2. neg-mul-1 93.1%

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

   8. Simplified 93.1%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.1%

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

      3. sqr-neg 96.1%

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

      4. sqrt-unprod 93.1%

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

      5. add-sqr-sqrt 93.1%

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

      6. *-un-lft-identity 93.1%

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

   10. Applied egg-rr 93.1%

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

       1. *-lft-identity 93.1%

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

   12. Simplified 93.1%

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

       1. clear-num 100.0%

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

       2. inv-pow 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   16. Simplified 100.0%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999933815813 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq 9.999999933815813 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -2000.0)
     (/ 1.0 (* x (/ 2.0 x)))
     (if (<= t_0 9.999999933815813e+36)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= -2000.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (t_0 <= 9.999999933815813e+36f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / (2.0f + (x / s)));
	} 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 <= (-2000.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (t_0 <= 9.999999933815813e+36) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (2.0e0 + (x / s)))
    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(-2000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(9.999999933815813e+36))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(2.0) + Float32(x / s))));
	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(-2000.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (t_0 <= single(9.999999933815813e+36))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / (single(2.0) + (x / s)));
	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) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. neg-mul-1 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. sub-neg 5.1%

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

      2. associate-*r/ 5.1%

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

      3. metadata-eval 5.1%

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

      4. distribute-neg-frac 5.1%

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

      5. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -2e3 < (/.f32 (neg.f32 x) s) < 9.99999993e36

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 52.4%

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

      1. neg-mul-1 52.4%

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

      2. unsub-neg 52.4%

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

   5. Simplified 52.4%

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

      1. clear-num 52.4%

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

      2. associate-/r/ 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. associate-*l/ 52.4%

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

      2. *-un-lft-identity 52.4%

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

      3. clear-num 52.4%

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

   9. Applied egg-rr 52.4%

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

       1. sub-neg 52.4%

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

       2. flip-+ 75.2%

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

       3. metadata-eval 75.2%

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

       4. clear-num 75.2%

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

       5. clear-num 75.2%

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

       6. clear-num 75.2%

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

   11. Applied egg-rr 75.2%

       \[\leadsto \frac{1}{\color{blue}{\frac{4 - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}}} \]
   12. Step-by-step derivation

       1. sqr-neg 75.2%

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

       2. sub-neg 75.2%

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

       3. remove-double-neg 75.2%

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

   13. Simplified 75.2%

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

   if 9.99999993e36 < (/.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. neg-mul-1 100.0%

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

      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 93.1%

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

      1. associate-*r/ 93.1%

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

      2. neg-mul-1 93.1%

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

   8. Simplified 93.1%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 96.1%

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

      3. sqr-neg 96.1%

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

      4. sqrt-unprod 93.1%

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

      5. add-sqr-sqrt 93.1%

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

      6. *-un-lft-identity 93.1%

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

   10. Applied egg-rr 93.1%

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

       1. *-lft-identity 93.1%

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

   12. Simplified 93.1%

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

       1. clear-num 100.0%

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

       2. inv-pow 100.0%

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

   14. Applied egg-rr 100.0%

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

       1. unpow-1 100.0%

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

   16. Simplified 100.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 9.999999933815813 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.05000000074505806)
   (/ (/ 1.0 x) (/ 2.0 x))
   (/ -1.0 (* x (* (/ 1.0 (* x s)) (- x (* s 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.05000000074505806f) {
		tmp = (1.0f / x) / (2.0f / x);
	} else {
		tmp = -1.0f / (x * ((1.0f / (x * s)) * (x - (s * 2.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 0.05000000074505806e0) then
        tmp = (1.0e0 / x) / (2.0e0 / x)
    else
        tmp = (-1.0e0) / (x * ((1.0e0 / (x * s)) * (x - (s * 2.0e0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(2.0) / x));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(Float32(1.0) / Float32(x * s)) * Float32(x - Float32(s * Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.05000000074505806))
		tmp = (single(1.0) / x) / (single(2.0) / x);
	else
		tmp = single(-1.0) / (x * ((single(1.0) / (x * s)) * (x - (s * single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.4%

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

      1. neg-mul-1 45.4%

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

      2. unsub-neg 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 45.2%

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

      1. sub-neg 45.2%

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

      2. associate-*r/ 45.2%

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

      3. metadata-eval 45.2%

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

      4. distribute-neg-frac 45.2%

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

      5. metadata-eval 45.2%

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

   8. Simplified 45.2%

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

   9. Taylor expanded in x around 0 54.9%

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

       1. inv-pow 54.9%

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

       2. unpow-prod-down 55.0%

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

       3. inv-pow 55.0%

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

   11. Applied egg-rr 55.0%

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

       1. unpow-1 55.0%

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

       2. associate-*r/ 55.1%

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

       3. *-rgt-identity 55.1%

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

   13. Simplified 55.1%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 43.8%

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

      1. neg-mul-1 43.8%

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

      2. unsub-neg 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around inf 43.8%

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

      1. sub-neg 43.8%

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

      2. associate-*r/ 43.8%

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

      3. metadata-eval 43.8%

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

      4. distribute-neg-frac 43.8%

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

      5. metadata-eval 43.8%

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

   8. Simplified 43.8%

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

      1. frac-add 53.1%

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

      2. div-inv 56.4%

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

      3. fma-define 56.4%

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

   10. Applied egg-rr 56.4%

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

       1. *-commutative 56.4%

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

       2. neg-mul-1 56.4%

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

       3. fmm-def 56.4%

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

       4. *-commutative 56.4%

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

   12. Simplified 56.4%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.05000000074505806)
   (/ (/ 1.0 x) (/ 2.0 x))
   (/ -1.0 (* x (/ (- x (* s 2.0)) (* x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.05000000074505806f) {
		tmp = (1.0f / x) / (2.0f / x);
	} else {
		tmp = -1.0f / (x * ((x - (s * 2.0f)) / (x * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 0.05000000074505806e0) then
        tmp = (1.0e0 / x) / (2.0e0 / x)
    else
        tmp = (-1.0e0) / (x * ((x - (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(0.05000000074505806))
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(2.0) / x));
	else
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(x - Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(0.05000000074505806))
		tmp = (single(1.0) / x) / (single(2.0) / x);
	else
		tmp = single(-1.0) / (x * ((x - (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.0500000007

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.4%

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

      1. neg-mul-1 45.4%

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

      2. unsub-neg 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around inf 45.2%

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

      1. sub-neg 45.2%

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

      2. associate-*r/ 45.2%

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

      3. metadata-eval 45.2%

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

      4. distribute-neg-frac 45.2%

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

      5. metadata-eval 45.2%

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

   8. Simplified 45.2%

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

   9. Taylor expanded in x around 0 54.9%

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

       1. inv-pow 54.9%

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

       2. unpow-prod-down 55.0%

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

       3. inv-pow 55.0%

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

   11. Applied egg-rr 55.0%

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

       1. unpow-1 55.0%

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

       2. associate-*r/ 55.1%

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

       3. *-rgt-identity 55.1%

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

   13. Simplified 55.1%

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

   if 0.0500000007 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 43.8%

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

      1. neg-mul-1 43.8%

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

      2. unsub-neg 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around inf 43.8%

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

      1. sub-neg 43.8%

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

      2. associate-*r/ 43.8%

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

      3. metadata-eval 43.8%

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

      4. distribute-neg-frac 43.8%

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

      5. metadata-eval 43.8%

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

   8. Simplified 43.8%

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

      1. frac-add 53.1%

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

      2. div-inv 56.4%

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

      3. fma-define 56.4%

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

   10. Applied egg-rr 56.4%

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

       1. *-commutative 56.4%

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

       2. neg-mul-1 56.4%

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

       3. fmm-def 56.4%

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

       4. *-commutative 56.4%

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

   12. Simplified 56.4%

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

       1. un-div-inv 53.1%

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

       2. *-commutative 53.1%

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

       3. *-commutative 53.1%

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

   14. Applied egg-rr 53.1%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.05000000074505806:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \frac{x - s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 40:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot 2}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 40.0)
   (/ (/ 1.0 x) (/ 2.0 x))
   (/ 1.0 (* x (/ (+ x (* s 2.0)) (* x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 40.0f) {
		tmp = (1.0f / x) / (2.0f / x);
	} else {
		tmp = 1.0f / (x * ((x + (s * 2.0f)) / (x * s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= 40.0e0) then
        tmp = (1.0e0 / x) / (2.0e0 / x)
    else
        tmp = 1.0e0 / (x * ((x + (s * 2.0e0)) / (x * s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(40.0))
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(2.0) / x));
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x + Float32(s * Float32(2.0))) / Float32(x * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(40.0))
		tmp = (single(1.0) / x) / (single(2.0) / x);
	else
		tmp = single(1.0) / (x * ((x + (s * single(2.0))) / (x * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 40

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.0%

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

      1. neg-mul-1 45.0%

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

      2. unsub-neg 45.0%

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

   5. Simplified 45.0%

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

   6. Taylor expanded in x around inf 44.9%

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

      1. sub-neg 44.9%

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

      2. associate-*r/ 44.9%

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

      3. metadata-eval 44.9%

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

      4. distribute-neg-frac 44.9%

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

      5. metadata-eval 44.9%

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

   8. Simplified 44.9%

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

   9. Taylor expanded in x around 0 54.3%

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

       1. inv-pow 54.3%

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

       2. unpow-prod-down 54.4%

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

       3. inv-pow 54.4%

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

   11. Applied egg-rr 54.4%

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

       1. unpow-1 54.4%

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

       2. associate-*r/ 54.4%

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

       3. *-rgt-identity 54.4%

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

   13. Simplified 54.4%

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

   if 40 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 44.3%

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

      1. neg-mul-1 44.3%

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

      2. unsub-neg 44.3%

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

   5. Simplified 44.3%

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

   6. Taylor expanded in x around inf 44.3%

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

      1. sub-neg 44.3%

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

      2. associate-*r/ 44.3%

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

      3. metadata-eval 44.3%

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

      4. distribute-neg-frac 44.3%

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

      5. metadata-eval 44.3%

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

   8. Simplified 44.3%

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

      1. +-commutative 44.3%

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

      2. clear-num 44.3%

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

      3. frac-add 53.9%

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

      4. *-un-lft-identity 53.9%

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

      5. frac-2neg 53.9%

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

      6. metadata-eval 53.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 53.9%

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

      9. sqr-neg 53.9%

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

      10. sqrt-unprod 53.9%

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

      11. add-sqr-sqrt 53.9%

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

      12. /-rgt-identity 53.9%

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

      13. frac-2neg 53.9%

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

      14. metadata-eval 53.9%

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

      15. add-sqr-sqrt -0.0%

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

      16. sqrt-unprod 67.1%

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

      17. sqr-neg 67.1%

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

      18. sqrt-unprod 53.9%

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

      19. add-sqr-sqrt 53.9%

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

      20. /-rgt-identity 53.9%

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

   10. Applied egg-rr 53.9%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 40:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot 2}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(s \cdot 2 - x\right)}{x \cdot s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.9999999996399175e-23)
   (/ (/ 1.0 x) (/ 2.0 x))
   (/ 1.0 (/ (* x (- (* s 2.0) x)) (* x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.9999999996399175e-23f) {
		tmp = (1.0f / x) / (2.0f / x);
	} else {
		tmp = 1.0f / ((x * ((s * 2.0f) - x)) / (x * s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 1.9999999996399175e-23) then
        tmp = (1.0e0 / x) / (2.0e0 / x)
    else
        tmp = 1.0e0 / ((x * ((s * 2.0e0) - x)) / (x * s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(2.0) / x));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(s * Float32(2.0)) - x)) / Float32(x * s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.9999999996399175e-23))
		tmp = (single(1.0) / x) / (single(2.0) / x);
	else
		tmp = single(1.0) / ((x * ((s * single(2.0)) - x)) / (x * s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 41.1%

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

      1. neg-mul-1 41.1%

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

      2. unsub-neg 41.1%

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

   5. Simplified 41.1%

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

   6. Taylor expanded in x around inf 41.0%

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

      1. sub-neg 41.0%

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

      2. associate-*r/ 41.0%

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

      3. metadata-eval 41.0%

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

      4. distribute-neg-frac 41.0%

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

      5. metadata-eval 41.0%

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

   8. Simplified 41.0%

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

   9. Taylor expanded in x around 0 52.1%

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

       1. inv-pow 52.1%

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

       2. unpow-prod-down 52.2%

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

       3. inv-pow 52.2%

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

   11. Applied egg-rr 52.2%

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

       1. unpow-1 52.2%

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

       2. associate-*r/ 52.3%

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

       3. *-rgt-identity 52.3%

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

   13. Simplified 52.3%

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

   if 2e-23 < (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 49.2%

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

      1. neg-mul-1 49.2%

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

      2. unsub-neg 49.2%

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

   5. Simplified 49.2%

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

   6. Taylor expanded in x around inf 49.2%

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

      1. sub-neg 49.2%

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

      2. associate-*r/ 49.2%

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

      3. metadata-eval 49.2%

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

      4. distribute-neg-frac 49.2%

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

      5. metadata-eval 49.2%

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

   8. Simplified 49.2%

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

      1. frac-add 55.3%

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

      2. div-inv 57.2%

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

      3. fma-define 57.2%

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

   10. Applied egg-rr 57.2%

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

       1. *-commutative 57.2%

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

       2. neg-mul-1 57.2%

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

       3. fmm-def 57.2%

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

       4. *-commutative 57.2%

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

   12. Simplified 57.2%

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

       1. *-commutative 57.2%

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

       2. un-div-inv 55.3%

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

       3. associate-*l/ 59.0%

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

       4. *-commutative 59.0%

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

       5. *-commutative 59.0%

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

   14. Applied egg-rr 59.0%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(s \cdot 2 - x\right)}{x \cdot s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{1}{x}}{s \cdot 2 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.4999999523982838e-21)
   (/ (/ 1.0 x) (/ 2.0 x))
   (* (* x s) (/ (/ 1.0 x) (- (* s 2.0) x)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.4999999523982838e-21f) {
		tmp = (1.0f / x) / (2.0f / x);
	} else {
		tmp = (x * s) * ((1.0f / x) / ((s * 2.0f) - 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 <= 1.4999999523982838e-21) then
        tmp = (1.0e0 / x) / (2.0e0 / x)
    else
        tmp = (x * s) * ((1.0e0 / x) / ((s * 2.0e0) - x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.4999999523982838e-21))
		tmp = Float32(Float32(Float32(1.0) / x) / Float32(Float32(2.0) / x));
	else
		tmp = Float32(Float32(x * s) * Float32(Float32(Float32(1.0) / x) / Float32(Float32(s * Float32(2.0)) - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.4999999523982838e-21))
		tmp = (single(1.0) / x) / (single(2.0) / x);
	else
		tmp = (x * s) * ((single(1.0) / x) / ((s * single(2.0)) - x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.5e-21

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 42.1%

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

      1. neg-mul-1 42.1%

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

      2. unsub-neg 42.1%

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

   5. Simplified 42.1%

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

   6. Taylor expanded in x around inf 42.0%

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

      1. sub-neg 42.0%

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

      2. associate-*r/ 42.0%

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

      3. metadata-eval 42.0%

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

      4. distribute-neg-frac 42.0%

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

      5. metadata-eval 42.0%

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

   8. Simplified 42.0%

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

   9. Taylor expanded in x around 0 51.9%

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

       1. inv-pow 51.9%

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

       2. unpow-prod-down 52.0%

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

       3. inv-pow 52.0%

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

   11. Applied egg-rr 52.0%

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

       1. unpow-1 52.0%

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

       2. associate-*r/ 52.0%

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

       3. *-rgt-identity 52.0%

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

   13. Simplified 52.0%

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

   if 1.5e-21 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 48.3%

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

      1. neg-mul-1 48.3%

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

      2. unsub-neg 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in x around inf 48.3%

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

      1. sub-neg 48.3%

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

      2. associate-*r/ 48.3%

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

      3. metadata-eval 48.3%

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

      4. distribute-neg-frac 48.3%

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

      5. metadata-eval 48.3%

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

   8. Simplified 48.3%

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

      1. frac-add 54.2%

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

      2. div-inv 57.4%

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

      3. fma-define 57.4%

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

   10. Applied egg-rr 57.4%

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

       1. *-commutative 57.4%

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

       2. neg-mul-1 57.4%

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

       3. fmm-def 57.4%

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

       4. *-commutative 57.4%

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

   12. Simplified 57.4%

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

       1. associate-/r* 54.9%

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

       2. *-un-lft-identity 54.9%

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

       3. *-commutative 54.9%

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

       4. times-frac 56.5%

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

       5. clear-num 53.3%

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

       6. /-rgt-identity 53.3%

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

       7. *-commutative 53.3%

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

       8. *-commutative 53.3%

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

   14. Applied egg-rr 53.3%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.4999999523982838 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot s\right) \cdot \frac{\frac{1}{x}}{s \cdot 2 - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 - \frac{x}{s}\right) + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -2000.0)
   (/ 1.0 (* x (/ 2.0 x)))
   (/ 1.0 (+ (- 3.0 (/ x s)) -1.0))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2000.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else {
		tmp = 1.0f / ((3.0f - (x / s)) + -1.0f);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((x / -s) <= (-2000.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else
        tmp = 1.0e0 / ((3.0e0 - (x / s)) + (-1.0e0))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-2000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(3.0) - Float32(x / s)) + Float32(-1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((x / -s) <= single(-2000.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	else
		tmp = single(1.0) / ((single(3.0) - (x / s)) + single(-1.0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. neg-mul-1 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. sub-neg 5.1%

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

      2. associate-*r/ 5.1%

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

      3. metadata-eval 5.1%

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

      4. distribute-neg-frac 5.1%

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

      5. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -2e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. neg-mul-1 63.4%

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

      2. unsub-neg 63.4%

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

   5. Simplified 63.4%

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

      1. expm1-log1p-u 63.3%

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

   7. Applied egg-rr 63.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 63.3%

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

      2. sub-neg 63.3%

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

      3. log1p-undefine 63.2%

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

      4. rem-exp-log 63.4%

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

      5. associate-+r- 63.4%

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

      6. metadata-eval 63.4%

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

      7. metadata-eval 63.4%

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

   9. Simplified 63.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(3 - \frac{x}{s}\right) + -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -2000.0) (/ 1.0 (* x (/ 2.0 x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -2000.0f) {
		tmp = 1.0f / (x * (2.0f / 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 / -s) <= (-2000.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / 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(-s)) <= Float32(-2000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / 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 / -s) <= single(-2000.0))
		tmp = single(1.0) / (x * (single(2.0) / 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 (/.f32 (neg.f32 x) s) < -2e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.1%

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

      1. neg-mul-1 5.1%

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

      2. unsub-neg 5.1%

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

   5. Simplified 5.1%

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

   6. Taylor expanded in x around inf 5.1%

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

      1. sub-neg 5.1%

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

      2. associate-*r/ 5.1%

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

      3. metadata-eval 5.1%

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

      4. distribute-neg-frac 5.1%

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

      5. metadata-eval 5.1%

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

   8. Simplified 5.1%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -2e3 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 63.4%

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

      1. neg-mul-1 63.4%

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

      2. unsub-neg 63.4%

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

   5. Simplified 63.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -2000:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.1% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.4000000059604645f) {
		tmp = 0.5f;
	} 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 / -s) <= 0.4000000059604645e0) then
        tmp = 0.5e0
    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(-s)) <= Float32(0.4000000059604645))
		tmp = Float32(0.5);
	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 / -s) <= single(0.4000000059604645))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x * (single(1.0) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.9%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 43.8%

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

      1. neg-mul-1 43.8%

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

      2. unsub-neg 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around inf 43.8%

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

      1. sub-neg 43.8%

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

      2. associate-*r/ 43.8%

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

      3. metadata-eval 43.8%

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

      4. distribute-neg-frac 43.8%

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

      5. metadata-eval 43.8%

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

   8. Simplified 43.8%

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

   9. Taylor expanded in x around inf 43.8%

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

4. Final simplification 50.2%

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

Alternative 12: 48.1% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.9%

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

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

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 43.8%

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

      1. neg-mul-1 43.8%

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

      2. unsub-neg 43.8%

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

   5. Simplified 43.8%

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

   6. Taylor expanded in x around inf 43.8%

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

      1. associate-*r/ 43.8%

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

      2. neg-mul-1 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 50.2%

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

Alternative 13: 47.8% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.199999957179898 \cdot 10^{-7}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.199999957179898e-7) 0.5 (/ 1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.199999957179898e-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 <= 1.199999957179898e-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(1.199999957179898e-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(1.199999957179898e-7))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1.19999996e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.9%

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

   if 1.19999996e-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 54.8%

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

      1. neg-mul-1 54.8%

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

      2. unsub-neg 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in x around inf 51.2%

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

      1. associate-*r/ 51.2%

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

      2. neg-mul-1 51.2%

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

   8. Simplified 51.2%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 67.1%

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

      3. sqr-neg 67.1%

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

      4. sqrt-unprod 50.9%

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

      5. add-sqr-sqrt 50.9%

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

      6. *-un-lft-identity 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. *-lft-identity 50.9%

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

   12. Simplified 50.9%

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

       1. clear-num 54.3%

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

       2. inv-pow 54.3%

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

   14. Applied egg-rr 54.3%

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

       1. unpow-1 54.3%

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

   16. Simplified 54.3%

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

Alternative 14: 46.4% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999997e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 52.2%

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

      1. neg-mul-1 52.2%

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

      2. unsub-neg 52.2%

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

   5. Simplified 52.2%

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

   6. Taylor expanded in x around inf 48.8%

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

      1. associate-*r/ 48.8%

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

      2. neg-mul-1 48.8%

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

   8. Simplified 48.8%

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

   if -4.99999997e-9 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 49.1%

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

Alternative 15: 46.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.199999957179898 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.199999957179898e-7) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.199999957179898e-7f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.199999957179898e-7)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.199999957179898e-7))
		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(-1.199999957179898e-7))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.19999996e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.8%

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

      1. neg-mul-1 54.8%

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

      2. unsub-neg 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in x around inf 51.2%

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

      1. associate-*r/ 51.2%

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

      2. neg-mul-1 51.2%

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

   8. Simplified 51.2%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 67.1%

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

      3. sqr-neg 67.1%

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

      4. sqrt-unprod 50.9%

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

      5. add-sqr-sqrt 50.9%

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

      6. *-un-lft-identity 50.9%

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

   10. Applied egg-rr 50.9%

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

       1. *-lft-identity 50.9%

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

   12. Simplified 50.9%

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

   if -1.19999996e-7 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 47.9%

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

Alternative 16: 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 34.5%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))