Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 13

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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 0.007000000216066837:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 100000000376832:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \mathbf{elif}\;t\_0 \leq 1.9999999889914546 \cdot 10^{+33}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 0.007000000216066837)
     0.5
     (if (<= t_0 100000000376832.0)
       (/ -1.0 (* x (* (/ 1.0 (* x s)) (- x (* s 2.0)))))
       (if (<= t_0 1.9999999889914546e+33)
         (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
         (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x)))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 0.007000000216066837f) {
		tmp = 0.5f;
	} else if (t_0 <= 100000000376832.0f) {
		tmp = -1.0f / (x * ((1.0f / (x * s)) * (x - (s * 2.0f))));
	} else if (t_0 <= 1.9999999889914546e+33f) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	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 <= 0.007000000216066837e0) then
        tmp = 0.5e0
    else if (t_0 <= 100000000376832.0e0) then
        tmp = (-1.0e0) / (x * ((1.0e0 / (x * s)) * (x - (s * 2.0e0))))
    else if (t_0 <= 1.9999999889914546e+33) then
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (x / s))
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    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(0.007000000216066837))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(100000000376832.0))
		tmp = Float32(Float32(-1.0) / Float32(x * Float32(Float32(Float32(1.0) / Float32(x * s)) * Float32(x - Float32(s * Float32(2.0))))));
	elseif (t_0 <= Float32(1.9999999889914546e+33))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(0.007000000216066837))
		tmp = single(0.5);
	elseif (t_0 <= single(100000000376832.0))
		tmp = single(-1.0) / (x * ((single(1.0) / (x * s)) * (x - (s * single(2.0)))));
	elseif (t_0 <= single(1.9999999889914546e+33))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (x / s));
	else
		tmp = single(1.0) / ((x * (single(2.0) + (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.00700000022

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.8%

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

   if 0.00700000022 < (/.f32 (neg.f32 x) s) < 1e14

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 15.2%

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

      1. mul-1-neg 15.2%

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

      2. unsub-neg 15.2%

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

   5. Simplified 15.2%

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

   6. Taylor expanded in x around inf 15.2%

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

      1. sub-neg 15.2%

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

      2. associate-*r/ 15.2%

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

      3. metadata-eval 15.2%

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

      4. distribute-neg-frac 15.2%

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

      5. metadata-eval 15.2%

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

   8. Simplified 15.2%

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

      1. frac-add 48.8%

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

      2. div-inv 56.5%

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

      3. fma-define 56.5%

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

   10. Applied egg-rr 56.5%

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

       1. *-commutative 56.5%

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

       2. mul-1-neg 56.5%

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

       3. fmm-def 56.5%

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

       4. *-commutative 56.5%

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

   12. Simplified 56.5%

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

   if 1e14 < (/.f32 (neg.f32 x) s) < 1.99999999e33

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 11.3%

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

      1. mul-1-neg 11.3%

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

      2. unsub-neg 11.3%

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

   5. Simplified 11.3%

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

      1. sub-neg 11.3%

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

      2. flip-+ 89.6%

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

      3. metadata-eval 89.6%

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

      4. distribute-neg-frac2 89.6%

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

      5. distribute-neg-frac2 89.6%

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

      6. distribute-neg-frac2 89.6%

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

   7. Applied egg-rr 89.6%

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

   8. Taylor expanded in x around inf 89.6%

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

   if 1.99999999e33 < (/.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 94.0%

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

      1. mul-1-neg 94.0%

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

      2. unsub-neg 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in x around inf 94.0%

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

      1. sub-neg 94.0%

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

      2. associate-*r/ 94.0%

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

      3. metadata-eval 94.0%

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

      4. distribute-neg-frac 94.0%

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

      5. metadata-eval 94.0%

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

   8. Simplified 94.0%

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

   9. Taylor expanded in x around 0 94.0%

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

       1. mul-1-neg 94.0%

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

       2. sub-neg 94.0%

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

   11. Simplified 94.0%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. div-inv 100.0%

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

       4. distribute-rgt-neg-in 100.0%

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

       5. mul-1-neg 100.0%

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

       6. div-inv 100.0%

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

       7. add-sqr-sqrt -0.0%

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

       8. sqrt-unprod 100.0%

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

       9. frac-times 100.0%

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

       10. metadata-eval 100.0%

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

       11. metadata-eval 100.0%

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

       12. frac-times 100.0%

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

       13. sqrt-unprod 100.0%

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

       14. add-sqr-sqrt 100.0%

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

       15. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.007000000216066837:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 100000000376832:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999889914546 \cdot 10^{+33}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ t_1 := 2 + \frac{x}{s}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999889914546 \cdot 10^{+33}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s \cdot \frac{s}{x}} - 4}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot t\_1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))) (t_1 (+ 2.0 (/ x s))))
   (if (<= t_0 -5.0)
     (/ 1.0 (* x (/ 2.0 x)))
     (if (<= t_0 1.9999999889914546e+33)
       (/ -1.0 (/ (- (/ x (* s (/ s x))) 4.0) t_1))
       (/ 1.0 (/ (* x t_1) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float t_1 = 2.0f + (x / s);
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f / (x * (2.0f / x));
	} else if (t_0 <= 1.9999999889914546e+33f) {
		tmp = -1.0f / (((x / (s * (s / x))) - 4.0f) / t_1);
	} else {
		tmp = 1.0f / ((x * t_1) / x);
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = x / -s
    t_1 = 2.0e0 + (x / s)
    if (t_0 <= (-5.0e0)) then
        tmp = 1.0e0 / (x * (2.0e0 / x))
    else if (t_0 <= 1.9999999889914546e+33) then
        tmp = (-1.0e0) / (((x / (s * (s / x))) - 4.0e0) / t_1)
    else
        tmp = 1.0e0 / ((x * t_1) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	t_1 = Float32(Float32(2.0) + Float32(x / s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(2.0) / x)));
	elseif (t_0 <= Float32(1.9999999889914546e+33))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(x / Float32(s * Float32(s / x))) - Float32(4.0)) / t_1));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * t_1) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	t_1 = single(2.0) + (x / s);
	tmp = single(0.0);
	if (t_0 <= single(-5.0))
		tmp = single(1.0) / (x * (single(2.0) / x));
	elseif (t_0 <= single(1.9999999889914546e+33))
		tmp = single(-1.0) / (((x / (s * (s / x))) - single(4.0)) / t_1);
	else
		tmp = single(1.0) / ((x * t_1) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. sub-neg 5.3%

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

      2. associate-*r/ 5.3%

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

      3. metadata-eval 5.3%

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

      4. distribute-neg-frac 5.3%

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

      5. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1.99999999e33

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. unsub-neg 54.5%

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

   5. Simplified 54.5%

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

      1. sub-neg 54.5%

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

      2. flip-+ 77.5%

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

      3. metadata-eval 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. distribute-neg-frac2 77.5%

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

      6. distribute-neg-frac2 77.5%

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

   7. Applied egg-rr 77.5%

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

      1. clear-num 77.5%

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

      2. frac-times 78.3%

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

      3. *-un-lft-identity 78.3%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 81.2%

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

      6. sqr-neg 81.2%

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

      7. sqrt-unprod 78.3%

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

      8. add-sqr-sqrt 78.3%

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

      9. add-sqr-sqrt -0.0%

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

      10. sqrt-unprod 81.2%

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

      11. sqr-neg 81.2%

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

      12. sqrt-unprod 78.3%

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

      13. add-sqr-sqrt 78.3%

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

   9. Applied egg-rr 78.3%

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

   if 1.99999999e33 < (/.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 94.0%

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

      1. mul-1-neg 94.0%

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

      2. unsub-neg 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in x around inf 94.0%

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

      1. sub-neg 94.0%

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

      2. associate-*r/ 94.0%

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

      3. metadata-eval 94.0%

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

      4. distribute-neg-frac 94.0%

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

      5. metadata-eval 94.0%

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

   8. Simplified 94.0%

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

   9. Taylor expanded in x around 0 94.0%

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

       1. mul-1-neg 94.0%

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

       2. sub-neg 94.0%

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

   11. Simplified 94.0%

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

       1. associate-*r/ 100.0%

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

       2. sub-neg 100.0%

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

       3. div-inv 100.0%

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

       4. distribute-rgt-neg-in 100.0%

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

       5. mul-1-neg 100.0%

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

       6. div-inv 100.0%

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

       7. add-sqr-sqrt -0.0%

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

       8. sqrt-unprod 100.0%

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

       9. frac-times 100.0%

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

       10. metadata-eval 100.0%

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

       11. metadata-eval 100.0%

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

       12. frac-times 100.0%

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

       13. sqrt-unprod 100.0%

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

       14. add-sqr-sqrt 100.0%

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

       15. div-inv 100.0%

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

   13. Applied egg-rr 100.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -5:\\ \;\;\;\;\frac{1}{x \cdot \frac{2}{x}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999889914546 \cdot 10^{+33}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s \cdot \frac{s}{x}} - 4}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot -2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 5.000000100204387e+20)
       (/ 1.0 (* x (/ (+ x (* s -2.0)) (* x s))))
       (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 5.000000100204387e+20f) {
		tmp = 1.0f / (x * ((x + (s * -2.0f)) / (x * s)));
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	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 <= 2.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 5.000000100204387e+20) then
        tmp = 1.0e0 / (x * ((x + (s * (-2.0e0))) / (x * s)))
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    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(2.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(5.000000100204387e+20))
		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(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(2.0))
		tmp = single(0.5);
	elseif (t_0 <= single(5.000000100204387e+20))
		tmp = single(1.0) / (x * ((x + (s * single(-2.0))) / (x * s)));
	else
		tmp = single(1.0) / ((x * (single(2.0) + (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.5%

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

   if 2 < (/.f32 (neg.f32 x) s) < 5.0000001e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. unsub-neg 7.7%

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

   5. Simplified 7.7%

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

   6. Taylor expanded in x around inf 7.7%

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

      1. sub-neg 7.7%

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

      2. associate-*r/ 7.7%

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

      3. metadata-eval 7.7%

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

      4. distribute-neg-frac 7.7%

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

      5. metadata-eval 7.7%

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

   8. Simplified 7.7%

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

      1. +-commutative 7.7%

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

      2. frac-2neg 7.7%

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

      3. metadata-eval 7.7%

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

      4. frac-2neg 7.7%

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

      5. frac-add 38.7%

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

      6. *-un-lft-identity 38.7%

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

      7. add-sqr-sqrt 38.7%

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

      8. sqrt-unprod 38.7%

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

      9. sqr-neg 38.7%

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

      10. sqrt-unprod -0.0%

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

      11. add-sqr-sqrt 38.7%

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

      12. add-sqr-sqrt -0.0%

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

      13. sqrt-unprod 38.7%

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

      14. sqr-neg 38.7%

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

      15. sqrt-unprod 38.7%

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

      16. add-sqr-sqrt 38.7%

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

      17. metadata-eval 38.7%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 52.2%

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

      20. sqr-neg 52.2%

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

      21. sqrt-unprod 38.7%

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

      22. add-sqr-sqrt 38.7%

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

      23. add-sqr-sqrt 38.7%

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

      24. sqrt-unprod 38.7%

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

      25. sqr-neg 38.7%

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

   10. Applied egg-rr 38.7%

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

   if 5.0000001e20 < (/.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 60.2%

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

      1. mul-1-neg 60.2%

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

      2. unsub-neg 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around inf 60.2%

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

      1. sub-neg 60.2%

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

      2. associate-*r/ 60.2%

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

      3. metadata-eval 60.2%

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

      4. distribute-neg-frac 60.2%

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

      5. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in x around 0 60.2%

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

       1. mul-1-neg 60.2%

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

       2. sub-neg 60.2%

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

   11. Simplified 60.2%

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

       1. associate-*r/ 76.6%

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

       2. sub-neg 76.6%

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

       3. div-inv 76.6%

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

       4. distribute-rgt-neg-in 76.6%

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

       5. mul-1-neg 76.6%

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

       6. div-inv 76.6%

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

       7. add-sqr-sqrt -0.0%

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

       8. sqrt-unprod 94.7%

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

       9. frac-times 94.7%

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

       10. metadata-eval 94.7%

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

       11. metadata-eval 94.7%

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

       12. frac-times 94.7%

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

       13. sqrt-unprod 76.6%

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

       14. add-sqr-sqrt 76.6%

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

       15. div-inv 76.6%

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

   13. Applied egg-rr 76.6%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{x \cdot \frac{x + s \cdot -2}{x \cdot s}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(x \cdot s\right) \cdot \frac{-1}{x}}{x - s \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 4.999999873689376e-5)
     0.5
     (if (<= t_0 5.000000100204387e+20)
       (/ (* (* x s) (/ -1.0 x)) (- x (* s 2.0)))
       (/ 1.0 (/ (* x (+ 2.0 (/ x s))) x))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 4.999999873689376e-5f) {
		tmp = 0.5f;
	} else if (t_0 <= 5.000000100204387e+20f) {
		tmp = ((x * s) * (-1.0f / x)) / (x - (s * 2.0f));
	} else {
		tmp = 1.0f / ((x * (2.0f + (x / s))) / x);
	}
	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 <= 4.999999873689376e-5) then
        tmp = 0.5e0
    else if (t_0 <= 5.000000100204387e+20) then
        tmp = ((x * s) * ((-1.0e0) / x)) / (x - (s * 2.0e0))
    else
        tmp = 1.0e0 / ((x * (2.0e0 + (x / s))) / x)
    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(4.999999873689376e-5))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(5.000000100204387e+20))
		tmp = Float32(Float32(Float32(x * s) * Float32(Float32(-1.0) / x)) / Float32(x - Float32(s * Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(Float32(2.0) + Float32(x / s))) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(4.999999873689376e-5))
		tmp = single(0.5);
	elseif (t_0 <= single(5.000000100204387e+20))
		tmp = ((x * s) * (single(-1.0) / x)) / (x - (s * single(2.0)));
	else
		tmp = single(1.0) / ((x * (single(2.0) + (x / s))) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 4.99999987e-5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.6%

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

   if 4.99999987e-5 < (/.f32 (neg.f32 x) s) < 5.0000001e20

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 22.3%

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

      1. mul-1-neg 22.3%

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

      2. unsub-neg 22.3%

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

   5. Simplified 22.3%

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

   6. Taylor expanded in x around inf 22.3%

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

      1. sub-neg 22.3%

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

      2. associate-*r/ 22.3%

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

      3. metadata-eval 22.3%

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

      4. distribute-neg-frac 22.3%

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

      5. metadata-eval 22.3%

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

   8. Simplified 22.3%

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

      1. frac-add 44.7%

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

      2. div-inv 51.6%

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

      3. fma-define 51.6%

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

   10. Applied egg-rr 51.6%

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

       1. *-commutative 51.6%

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

       2. mul-1-neg 51.6%

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

       3. fmm-def 51.6%

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

       4. *-commutative 51.6%

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

   12. Simplified 51.6%

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

       1. associate-/r* 51.7%

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

       2. *-un-lft-identity 51.7%

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

       3. *-commutative 51.7%

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

       4. times-frac 28.4%

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

       5. clear-num 18.2%

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

       6. /-rgt-identity 18.2%

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

       7. *-commutative 18.2%

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

   14. Applied egg-rr 18.2%

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

       1. associate-*r/ 44.8%

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

   16. Simplified 44.8%

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

   if 5.0000001e20 < (/.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 60.2%

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

      1. mul-1-neg 60.2%

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

      2. unsub-neg 60.2%

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

   5. Simplified 60.2%

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

   6. Taylor expanded in x around inf 60.2%

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

      1. sub-neg 60.2%

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

      2. associate-*r/ 60.2%

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

      3. metadata-eval 60.2%

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

      4. distribute-neg-frac 60.2%

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

      5. metadata-eval 60.2%

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

   8. Simplified 60.2%

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

   9. Taylor expanded in x around 0 60.2%

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

       1. mul-1-neg 60.2%

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

       2. sub-neg 60.2%

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

   11. Simplified 60.2%

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

       1. associate-*r/ 76.6%

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

       2. sub-neg 76.6%

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

       3. div-inv 76.6%

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

       4. distribute-rgt-neg-in 76.6%

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

       5. mul-1-neg 76.6%

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

       6. div-inv 76.6%

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

       7. add-sqr-sqrt -0.0%

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

       8. sqrt-unprod 94.7%

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

       9. frac-times 94.7%

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

       10. metadata-eval 94.7%

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

       11. metadata-eval 94.7%

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

       12. frac-times 94.7%

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

       13. sqrt-unprod 76.6%

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

       14. add-sqr-sqrt 76.6%

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

       15. div-inv 76.6%

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

   13. Applied egg-rr 76.6%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 4.999999873689376 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq 5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(x \cdot s\right) \cdot \frac{-1}{x}}{x - s \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(2 + \frac{x}{s}\right)}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.007000000216066837:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) 0.007000000216066837)
   0.5
   (/ -1.0 (* x (* (/ 1.0 (* x s)) (- x (* s 2.0)))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.007000000216066837f) {
		tmp = 0.5f;
	} 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) <= 0.007000000216066837e0) then
        tmp = 0.5e0
    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(0.007000000216066837))
		tmp = Float32(0.5);
	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(0.007000000216066837))
		tmp = single(0.5);
	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) < 0.00700000022

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.8%

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

   if 0.00700000022 < (/.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 45.7%

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

      1. mul-1-neg 45.7%

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

      2. unsub-neg 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in x around inf 45.7%

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

      1. sub-neg 45.7%

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

      2. associate-*r/ 45.7%

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

      3. metadata-eval 45.7%

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

      4. distribute-neg-frac 45.7%

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

      5. metadata-eval 45.7%

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

   8. Simplified 45.7%

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

      1. frac-add 54.7%

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

      2. div-inv 62.0%

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

      3. fma-define 62.0%

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

   10. Applied egg-rr 62.0%

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

       1. *-commutative 62.0%

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

       2. mul-1-neg 62.0%

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

       3. fmm-def 62.0%

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

       4. *-commutative 62.0%

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

   12. Simplified 62.0%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.007000000216066837:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot \left(\frac{1}{x \cdot s} \cdot \left(x - s \cdot 2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. unsub-neg 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in x around inf 45.3%

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

      1. sub-neg 45.3%

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

      2. associate-*r/ 45.3%

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

      3. metadata-eval 45.3%

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

      4. distribute-neg-frac 45.3%

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

      5. metadata-eval 45.3%

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

   8. Simplified 45.3%

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

   9. Taylor expanded in x around 0 45.3%

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

       1. mul-1-neg 45.3%

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

       2. sub-neg 45.3%

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

   11. Simplified 45.3%

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

       1. associate-*r/ 57.0%

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

       2. sub-neg 57.0%

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

       3. div-inv 57.0%

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

       4. distribute-rgt-neg-in 57.0%

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

       5. mul-1-neg 57.0%

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

       6. div-inv 57.0%

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

       7. add-sqr-sqrt -0.0%

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

       8. sqrt-unprod 83.6%

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

       9. frac-times 83.6%

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

       10. metadata-eval 83.6%

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

       11. metadata-eval 83.6%

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

       12. frac-times 83.6%

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

       13. sqrt-unprod 57.0%

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

       14. add-sqr-sqrt 57.0%

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

       15. div-inv 57.0%

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

   13. Applied egg-rr 57.0%

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

4. Final simplification 53.5%

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

Alternative 8: 52.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{s}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 3.999999999279835e-23) 0.5 (* x (/ s (* x (- (* s 2.0) x))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 3.999999999279835e-23f) {
		tmp = 0.5f;
	} else {
		tmp = x * (s / (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 <= 3.999999999279835e-23) then
        tmp = 0.5e0
    else
        tmp = x * (s / (x * ((s * 2.0e0) - x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(3.999999999279835e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(x * Float32(s / Float32(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(3.999999999279835e-23))
		tmp = single(0.5);
	else
		tmp = x * (s / (x * ((s * single(2.0)) - x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 4e-23

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 47.0%

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

   if 4e-23 < (neg.f32 x)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.2%

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

      1. mul-1-neg 54.2%

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

      2. unsub-neg 54.2%

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

   5. Simplified 54.2%

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

   6. Taylor expanded in x around inf 54.2%

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

      1. sub-neg 54.2%

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

      2. associate-*r/ 54.2%

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

      3. metadata-eval 54.2%

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

      4. distribute-neg-frac 54.2%

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

      5. metadata-eval 54.2%

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

   8. Simplified 54.2%

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

      1. frac-add 55.8%

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

      2. div-inv 61.5%

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

      3. fma-define 61.5%

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

   10. Applied egg-rr 61.5%

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

       1. *-commutative 61.5%

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

       2. mul-1-neg 61.5%

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

       3. fmm-def 61.5%

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

       4. *-commutative 61.5%

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

   12. Simplified 61.5%

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

       1. associate-/r* 56.8%

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

       2. *-un-lft-identity 56.8%

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

       3. *-commutative 56.8%

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

       4. times-frac 61.3%

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

       5. clear-num 54.3%

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

       6. /-rgt-identity 54.3%

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

       7. *-commutative 54.3%

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

   14. Applied egg-rr 54.3%

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

       1. associate-*l* 55.5%

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

       2. associate-/l/ 56.3%

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

       3. *-commutative 56.3%

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

       4. associate-*r/ 57.7%

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

       5. *-rgt-identity 57.7%

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

   16. Simplified 57.7%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{s}{x \cdot \left(s \cdot 2 - x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.8% accurate, 7.2× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -5.0) (/ 1.0 (* x (/ 2.0 x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -5.0f) {
		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) <= (-5.0e0)) 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(-5.0))
		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(-5.0))
		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) < -5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 5.3%

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

      1. mul-1-neg 5.3%

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

      2. unsub-neg 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in x around inf 5.3%

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

      1. sub-neg 5.3%

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

      2. associate-*r/ 5.3%

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

      3. metadata-eval 5.3%

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

      4. distribute-neg-frac 5.3%

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

      5. metadata-eval 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in x around 0 28.1%

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

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. unsub-neg 64.5%

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

   5. Simplified 64.5%

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

4. Final simplification 50.7%

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

Alternative 10: 48.2% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 51.5%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.3%

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

      1. mul-1-neg 45.3%

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

      2. unsub-neg 45.3%

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

   5. Simplified 45.3%

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

   6. Taylor expanded in x around inf 40.2%

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

      1. associate-*r/ 40.2%

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

      2. neg-mul-1 40.2%

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

   8. Simplified 40.2%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 65.4%

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

      3. sqr-neg 65.4%

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

      4. sqrt-unprod 40.2%

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

      5. add-sqr-sqrt 40.2%

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

      6. *-un-lft-identity 40.2%

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

   10. Applied egg-rr 40.2%

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

       1. *-lft-identity 40.2%

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

   12. Simplified 40.2%

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

       1. clear-num 45.3%

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

       2. inv-pow 45.3%

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

   14. Applied egg-rr 45.3%

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

       1. unpow-1 45.3%

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

   16. Simplified 45.3%

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

4. Final simplification 49.2%

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

Alternative 11: 46.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999987376214e-7) (* s (/ 1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999987376214e-7f) {
		tmp = s * (1.0f / x);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999987376214e-7)) then
        tmp = s * (1.0e0 / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999987376214e-7))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999987376214e-7))
		tmp = s * (single(1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around inf 47.7%

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

      1. associate-*r/ 47.7%

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

      2. neg-mul-1 47.7%

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

   8. Simplified 47.7%

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

      1. clear-num 54.0%

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

      2. distribute-frac-neg2 54.0%

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

      3. distribute-frac-neg 54.0%

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

      4. associate-/r/ 47.7%

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

      5. add-sqr-sqrt 47.7%

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

      6. sqrt-unprod 56.3%

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

      7. sqr-neg 56.3%

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

      8. sqrt-unprod -0.0%

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

      9. add-sqr-sqrt 47.7%

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

   10. Applied egg-rr 47.7%

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

   if -4.99999999e-7 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999987376214e-7) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999987376214e-7f) {
		tmp = s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999987376214e-7)) then
        tmp = s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999987376214e-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(-4.999999987376214e-7))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999999e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   5. Simplified 54.0%

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

   6. Taylor expanded in x around inf 47.7%

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

      1. associate-*r/ 47.7%

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

      2. neg-mul-1 47.7%

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

   8. Simplified 47.7%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-unprod 59.8%

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

      3. sqr-neg 59.8%

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

      4. sqrt-unprod 47.7%

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

      5. add-sqr-sqrt 47.7%

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

      6. *-un-lft-identity 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. *-lft-identity 47.7%

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

   12. Simplified 47.7%

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

   if -4.99999999e-7 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 46.9%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.4% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x s) :precision binary32 0.5)

C

float code(float x, float s) {
	return 0.5f;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.5e0
end function

Julia

function code(x, s)
	return Float32(0.5)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.5);
end

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 34.7%

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

4. Final simplification 34.7%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024089 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))