Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.8s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.8%

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

   3. div-inv 99.8%

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

   4. add-sqr-sqrt 50.3%

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

   5. sqrt-unprod 60.9%

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

   6. sqr-neg 60.9%

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

   7. sqrt-unprod 12.0%

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

   8. add-sqr-sqrt 25.5%

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

   9. div-inv 25.5%

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

   10. pow1 25.5%

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

   11. pow1 25.5%

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

   12. add-sqr-sqrt 25.5%

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

   13. pow2 25.5%

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

   14. metadata-eval 25.5%

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

   15. pow-flip 25.5%

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

3. Applied egg-rr 99.9%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

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

   4. add-exp-log 99.9%

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

   5. sqrt-pow2 99.9%

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

   6. metadata-eval 99.9%

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

   7. inv-pow 99.9%

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

   8. neg-log 99.9%

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

   9. add-log-exp 99.9%

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

   10. distribute-neg-frac 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 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(Float32(-x) / 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. Final simplification 99.8%

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

Alternative 3: 86.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (+ 1.0 (/ s x))
     (if (<= t_0 9.999999680285692e+37)
       (/ 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 <= -2.0f) {
		tmp = 1.0f + (s / x);
	} else if (t_0 <= 9.999999680285692e+37f) {
		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 <= (-2.0e0)) then
        tmp = 1.0e0 + (s / x)
    else if (t_0 <= 9.999999680285692e+37) 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(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (t_0 <= Float32(9.999999680285692e+37))
		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(-2.0))
		tmp = single(1.0) + (s / x);
	elseif (t_0 <= single(9.999999680285692e+37))
		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) < -2

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 93.9%

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

      1. frac-2neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. div-inv 93.9%

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

      4. +-commutative 93.9%

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

      5. distribute-neg-in 93.9%

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

      6. distribute-frac-neg 93.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 92.5%

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

      9. sqr-neg 92.5%

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

      10. sqrt-unprod 94.3%

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

      11. add-sqr-sqrt 94.3%

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

      12. metadata-eval 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. metadata-eval 94.3%

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

      2. sub-neg 94.3%

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

      3. associate-*r/ 94.3%

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

      4. metadata-eval 94.3%

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

      5. sub-neg 94.3%

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

      6. metadata-eval 94.3%

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

   8. Simplified 94.3%

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

   9. Taylor expanded in x around inf 94.3%

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

   if -2 < (/.f32 (neg.f32 x) s) < 9.99999968e37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 50.0%

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

      1. mul-1-neg 50.0%

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

      2. unsub-neg 50.0%

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

   4. Simplified 50.0%

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

      1. sub-neg 50.0%

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

      2. flip-+ 69.8%

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

      3. metadata-eval 69.8%

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

      4. distribute-neg-frac 69.8%

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

      5. distribute-neg-frac 69.8%

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

      6. distribute-neg-frac 69.8%

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

   6. Applied egg-rr 69.8%

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

      1. distribute-frac-neg 69.8%

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

      2. distribute-frac-neg 69.8%

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

      3. sqr-neg 69.8%

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

      4. clear-num 69.8%

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

      5. frac-times 73.4%

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

      6. *-un-lft-identity 73.4%

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

   8. Applied egg-rr 73.4%

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

   if 9.99999968e37 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   7. Simplified 100.0%

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

      1. associate-/r/ 88.6%

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

      2. add-sqr-sqrt 88.6%

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

      3. sqrt-unprod 97.3%

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

      4. sqr-neg 97.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 88.6%

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

   9. Applied egg-rr 88.6%

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

       1. associate-/r/ 100.0%

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

       2. frac-2neg 100.0%

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

       3. metadata-eval 100.0%

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

       4. distribute-frac-neg 100.0%

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

       5. add-sqr-sqrt 100.0%

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

       6. sqrt-unprod 100.0%

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

       7. sqr-neg 100.0%

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

       8. sqrt-unprod -0.0%

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

       9. add-sqr-sqrt 100.0%

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

   11. Applied egg-rr 100.0%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999680285692 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 4: 75.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -10:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -10.0)
     (+ 1.0 (/ s x))
     (if (<= t_0 0.10000000149011612)
       (+ 0.5 (/ (* x 0.25) s))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -10.0f) {
		tmp = 1.0f + (s / x);
	} else if (t_0 <= 0.10000000149011612f) {
		tmp = 0.5f + ((x * 0.25f) / 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 <= (-10.0e0)) then
        tmp = 1.0e0 + (s / x)
    else if (t_0 <= 0.10000000149011612e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-10.0))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / 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(-10.0))
		tmp = single(1.0) + (s / x);
	elseif (t_0 <= single(0.10000000149011612))
		tmp = single(0.5) + ((x * single(0.25)) / 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) < -10

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.5%

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

      1. frac-2neg 94.5%

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

      2. metadata-eval 94.5%

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

      3. div-inv 94.5%

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

      4. +-commutative 94.5%

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

      5. distribute-neg-in 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 93.2%

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

      9. sqr-neg 93.2%

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

      10. sqrt-unprod 95.0%

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

      11. add-sqr-sqrt 95.0%

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

      12. metadata-eval 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. metadata-eval 95.0%

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

      2. sub-neg 95.0%

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

      3. associate-*r/ 95.0%

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

      4. metadata-eval 95.0%

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

      5. sub-neg 95.0%

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

      6. metadata-eval 95.0%

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

   8. Simplified 95.0%

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

   9. Taylor expanded in x around inf 95.0%

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

   if -10 < (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 97.4%

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

      1. associate-*r/ 97.4%

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

   4. Simplified 97.4%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 38.5%

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

      1. mul-1-neg 38.5%

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

      2. unsub-neg 38.5%

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

   4. Simplified 38.5%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. mul-1-neg 38.5%

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

      2. distribute-frac-neg 38.5%

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

   7. Simplified 38.5%

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

      1. associate-/r/ 35.0%

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

      2. add-sqr-sqrt 35.0%

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

      3. sqrt-unprod 43.8%

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

      4. sqr-neg 43.8%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 34.8%

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

   9. Applied egg-rr 34.8%

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

       1. associate-/r/ 38.3%

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

       2. frac-2neg 38.3%

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

       3. metadata-eval 38.3%

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

       4. distribute-frac-neg 38.3%

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

       5. add-sqr-sqrt 38.3%

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

       6. sqrt-unprod 44.5%

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

       7. sqr-neg 44.5%

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

       8. sqrt-unprod -0.0%

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

       9. add-sqr-sqrt 38.5%

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

   11. Applied egg-rr 38.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -10:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 5: 73.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (+ 1.0 (/ s x))
     (if (<= t_0 0.10000000149011612) 0.5 (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f + (s / x);
	} else if (t_0 <= 0.10000000149011612f) {
		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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-2.0e0)) then
        tmp = 1.0e0 + (s / x)
    else if (t_0 <= 0.10000000149011612e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-2.0))
		tmp = Float32(Float32(1.0) + Float32(s / x));
	elseif (t_0 <= Float32(0.10000000149011612))
		tmp = Float32(0.5);
	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(-2.0))
		tmp = single(1.0) + (s / x);
	elseif (t_0 <= single(0.10000000149011612))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 93.9%

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

      1. frac-2neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. div-inv 93.9%

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

      4. +-commutative 93.9%

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

      5. distribute-neg-in 93.9%

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

      6. distribute-frac-neg 93.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 92.5%

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

      9. sqr-neg 92.5%

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

      10. sqrt-unprod 94.3%

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

      11. add-sqr-sqrt 94.3%

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

      12. metadata-eval 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. metadata-eval 94.3%

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

      2. sub-neg 94.3%

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

      3. associate-*r/ 94.3%

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

      4. metadata-eval 94.3%

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

      5. sub-neg 94.3%

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

      6. metadata-eval 94.3%

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

   8. Simplified 94.3%

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

   9. Taylor expanded in x around inf 94.3%

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

   if -2 < (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 91.6%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 38.5%

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

      1. mul-1-neg 38.5%

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

      2. unsub-neg 38.5%

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

   4. Simplified 38.5%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. mul-1-neg 38.5%

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

      2. distribute-frac-neg 38.5%

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

   7. Simplified 38.5%

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

      1. associate-/r/ 35.0%

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

      2. add-sqr-sqrt 35.0%

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

      3. sqrt-unprod 43.8%

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

      4. sqr-neg 43.8%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 34.8%

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

   9. Applied egg-rr 34.8%

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

       1. associate-/r/ 38.3%

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

       2. frac-2neg 38.3%

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

       3. metadata-eval 38.3%

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

       4. distribute-frac-neg 38.3%

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

       5. add-sqr-sqrt 38.3%

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

       6. sqrt-unprod 44.5%

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

       7. sqr-neg 44.5%

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

       8. sqrt-unprod -0.0%

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

       9. add-sqr-sqrt 38.5%

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

   11. Applied egg-rr 38.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1 + \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 6: 74.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 93.9%

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

      1. frac-2neg 93.9%

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

      2. metadata-eval 93.9%

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

      3. div-inv 93.9%

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

      4. +-commutative 93.9%

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

      5. distribute-neg-in 93.9%

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

      6. distribute-frac-neg 93.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 92.5%

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

      9. sqr-neg 92.5%

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

      10. sqrt-unprod 94.3%

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

      11. add-sqr-sqrt 94.3%

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

      12. metadata-eval 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. metadata-eval 94.3%

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

      2. sub-neg 94.3%

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

      3. associate-*r/ 94.3%

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

      4. metadata-eval 94.3%

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

      5. sub-neg 94.3%

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

      6. metadata-eval 94.3%

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

   8. Simplified 94.3%

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

   9. Taylor expanded in x around inf 94.3%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 59.8%

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

      1. mul-1-neg 59.8%

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

      2. unsub-neg 59.8%

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

   4. Simplified 59.8%

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

4. Final simplification 72.9%

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

Alternative 7: 67.8% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.000000026702864e-10)
   (/ (- s) x)
   (if (<= x 3.999999999279835e-23) 0.5 (+ 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.000000026702864e-10f) {
		tmp = -s / x;
	} else if (x <= 3.999999999279835e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f + (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-2.000000026702864e-10)) then
        tmp = -s / x
    else if (x <= 3.999999999279835e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 + (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.000000026702864e-10))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(3.999999999279835e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) + Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.000000026702864e-10))
		tmp = -s / x;
	elseif (x <= single(3.999999999279835e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0) + (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2.00000003e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 46.5%

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

      1. mul-1-neg 46.5%

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

      2. unsub-neg 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in x around inf 42.1%

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

      1. associate-*r/ 42.1%

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

      2. neg-mul-1 42.1%

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

   7. Simplified 42.1%

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

   if -2.00000003e-10 < x < 4e-23

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 66.5%

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

   if 4e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 94.1%

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

      1. frac-2neg 94.1%

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

      2. metadata-eval 94.1%

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

      3. div-inv 94.1%

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

      4. +-commutative 94.1%

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

      5. distribute-neg-in 94.1%

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

      6. distribute-frac-neg 94.1%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 93.0%

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

      9. sqr-neg 93.0%

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

      10. sqrt-unprod 93.0%

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

      11. add-sqr-sqrt 93.0%

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

      12. metadata-eval 93.0%

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

   6. Applied egg-rr 93.0%

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

      1. metadata-eval 93.0%

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

      2. sub-neg 93.0%

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

      3. associate-*r/ 93.0%

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

      4. metadata-eval 93.0%

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

      5. sub-neg 93.0%

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

      6. metadata-eval 93.0%

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

   8. Simplified 93.0%

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

   9. Taylor expanded in x around inf 88.5%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 3.999999999279835 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{s}{x}\\ \end{array} \]

Alternative 8: 46.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.000000026702864e-10) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.000000026702864e-10f) {
		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 <= (-2.000000026702864e-10)) 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(-2.000000026702864e-10))
		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(-2.000000026702864e-10))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000003e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 46.5%

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

      1. mul-1-neg 46.5%

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

      2. unsub-neg 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in x around inf 42.1%

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

      1. associate-*r/ 42.1%

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

      2. neg-mul-1 42.1%

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

   7. Simplified 42.1%

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

   if -2.00000003e-10 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 46.3% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.000000026702864e-10) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.000000026702864e-10f) {
		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 <= (-2.000000026702864e-10)) 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(-2.000000026702864e-10))
		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(-2.000000026702864e-10))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000003e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 46.5%

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

      1. mul-1-neg 46.5%

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

      2. unsub-neg 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in x around inf 46.5%

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

      1. mul-1-neg 46.5%

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

      2. distribute-frac-neg 46.5%

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

   7. Simplified 46.5%

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

      1. associate-/r/ 42.1%

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

      2. add-sqr-sqrt 42.1%

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

      3. sqrt-unprod 53.5%

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

      4. sqr-neg 53.5%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 42.0%

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

   9. Applied egg-rr 42.0%

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

   10. Taylor expanded in x around 0 42.0%

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

   if -2.00000003e-10 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 10: 35.5% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 34.0%

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

3. Final simplification 34.0%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023334 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))