Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.6s

Alternatives: 8

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. div-inv 99.8%

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

   2. exp-prod 81.2%

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

   3. neg-mul-1 81.2%

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

   4. exp-prod 81.2%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. add-exp-log 99.8%

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

   2. log-rec 99.8%

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

   3. log1p-udef 99.9%

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

   4. pow-exp 99.9%

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

   5. neg-mul-1 99.9%

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

   6. 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: 61.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.0020000000949949026:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{1}{x} \cdot \frac{s}{\frac{1}{s}}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 0.0020000000949949026)
     0.5
     (if (<= t_0 1.9999999867631625e+37)
       (/ 1.0 (/ (- 4.0 (/ x (* (/ 1.0 x) (/ s (/ 1.0 s))))) (+ 2.0 (/ x s))))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.0020000000949949026f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999867631625e+37f) {
		tmp = 1.0f / ((4.0f - (x / ((1.0f / x) * (s / (1.0f / s))))) / (2.0f + (x / s)));
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 0.0020000000949949026e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999867631625e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / ((1.0e0 / x) * (s / (1.0e0 / s))))) / (2.0e0 + (x / s)))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0020000000949949026))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999867631625e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(Float32(Float32(1.0) / x) * Float32(s / Float32(Float32(1.0) / s))))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.0020000000949949026))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999867631625e+37))
		tmp = single(1.0) / ((single(4.0) - (x / ((single(1.0) / x) * (s / (single(1.0) / s))))) / (single(2.0) + (x / s)));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 0.00200000009

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 52.3%

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

   if 0.00200000009 < (/.f32 (neg.f32 x) s) < 1.99999999e37

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 13.6%

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

      1. mul-1-neg 13.6%

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

      2. unsub-neg 13.6%

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

   4. Simplified 13.6%

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

      1. sub-neg 13.6%

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

      2. neg-mul-1 13.6%

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

      3. rem-log-exp 94.1%

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

      4. pow-exp 94.1%

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

      5. flip-+ 4.5%

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

      6. metadata-eval 4.5%

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

      7. pow-exp 4.5%

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

      8. rem-log-exp 4.5%

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

      9. pow-exp 4.5%

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

      10. rem-log-exp 5.9%

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

      11. neg-mul-1 5.9%

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

      12. distribute-neg-frac 5.9%

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

      13. neg-mul-1 5.9%

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

      14. distribute-neg-frac 5.9%

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

      15. pow-exp 5.9%

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

      16. rem-log-exp 55.5%

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

      17. neg-mul-1 55.5%

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

      18. distribute-neg-frac 55.5%

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

   6. Applied egg-rr 55.5%

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

      1. associate-*l/ 55.5%

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

      2. associate-/l* 56.9%

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

      3. add-sqr-sqrt 56.9%

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

      4. sqrt-unprod 56.8%

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

      5. sqr-neg 56.8%

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

      6. sqrt-unprod -0.0%

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

      7. add-sqr-sqrt 56.5%

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

      8. add-sqr-sqrt 56.5%

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

      9. sqrt-unprod 56.5%

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

      10. sqr-neg 56.5%

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

      11. sqrt-unprod -0.0%

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

      12. add-sqr-sqrt 56.9%

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

   8. Applied egg-rr 56.9%

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

      1. *-un-lft-identity 56.9%

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

      2. div-inv 56.9%

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

      3. times-frac 78.6%

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

   10. Applied egg-rr 78.6%

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

   if 1.99999999e37 < (/.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/ 90.9%

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

      2. add-sqr-sqrt 90.9%

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

      3. sqrt-unprod 96.7%

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

      4. sqr-neg 96.7%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. frac-2neg 90.9%

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

       2. metadata-eval 90.9%

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

       3. associate-/r/ 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod -0.0%

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

       8. 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 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.0020000000949949026:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{1}{x} \cdot \frac{s}{\frac{1}{s}}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 4: 59.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -4:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s}{\frac{x}{s}}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -4.0)
     0.5
     (if (<= t_0 1.9999999867631625e+37)
       (/ 1.0 (/ (- 4.0 (/ x (/ s (/ x s)))) (+ 2.0 (/ x s))))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -4.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999867631625e+37f) {
		tmp = 1.0f / ((4.0f - (x / (s / (x / s)))) / (2.0f + (x / s)));
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-4.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999867631625e+37) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s / (x / s)))) / (2.0e0 + (x / s)))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-4.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999867631625e+37))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s / Float32(x / s)))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-4.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999867631625e+37))
		tmp = single(1.0) / ((single(4.0) - (x / (s / (x / s)))) / (single(2.0) + (x / s)));
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

   if -4 < (/.f32 (neg.f32 x) s) < 1.99999999e37

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. mul-1-neg 52.6%

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

      2. unsub-neg 52.6%

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

   4. Simplified 52.6%

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

      1. sub-neg 52.6%

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

      2. neg-mul-1 52.6%

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

      3. rem-log-exp 95.7%

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

      4. pow-exp 95.7%

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

      5. flip-+ 47.8%

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

      6. metadata-eval 47.8%

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

      7. pow-exp 47.8%

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

      8. rem-log-exp 47.8%

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

      9. pow-exp 47.8%

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

      10. rem-log-exp 48.5%

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

      11. neg-mul-1 48.5%

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

      12. distribute-neg-frac 48.5%

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

      13. neg-mul-1 48.5%

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

      14. distribute-neg-frac 48.5%

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

      15. pow-exp 48.5%

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

      16. rem-log-exp 75.1%

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

      17. neg-mul-1 75.1%

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

      18. distribute-neg-frac 75.1%

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

   6. Applied egg-rr 75.1%

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

      1. associate-*l/ 75.1%

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

      2. associate-/l* 75.8%

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

      3. add-sqr-sqrt 56.7%

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

      4. sqrt-unprod 76.1%

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

      5. sqr-neg 76.1%

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

      6. sqrt-unprod 19.1%

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

      7. add-sqr-sqrt 75.6%

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

      8. add-sqr-sqrt 56.5%

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

      9. sqrt-unprod 75.9%

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

      10. sqr-neg 75.9%

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

      11. sqrt-unprod 19.1%

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

      12. add-sqr-sqrt 75.8%

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

   8. Applied egg-rr 75.8%

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

   if 1.99999999e37 < (/.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/ 90.9%

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

      2. add-sqr-sqrt 90.9%

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

      3. sqrt-unprod 96.7%

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

      4. sqr-neg 96.7%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. frac-2neg 90.9%

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

       2. metadata-eval 90.9%

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

       3. associate-/r/ 100.0%

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

       4. add-sqr-sqrt 100.0%

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

       5. sqrt-unprod 100.0%

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

       6. sqr-neg 100.0%

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

       7. sqrt-unprod -0.0%

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

       8. 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 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999867631625 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s}{\frac{x}{s}}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]

Alternative 5: 48.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -4.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -4.0f) {
		tmp = 0.5f;
	} 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) <= (-4.0e0)) then
        tmp = 0.5e0
    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(-4.0))
		tmp = Float32(0.5);
	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(-4.0))
		tmp = single(0.5);
	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) < -4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

   if -4 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. mul-1-neg 64.1%

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

      2. unsub-neg 64.1%

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

   4. Simplified 64.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 6: 47.3% accurate, 10.7× 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(Float32(-x) / 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 99.8%

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

   2. Taylor expanded in x around 0 51.9%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 45.4%

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

      1. mul-1-neg 45.4%

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

      2. unsub-neg 45.4%

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

   4. Simplified 45.4%

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

   5. Taylor expanded in x around inf 45.4%

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

      1. mul-1-neg 45.4%

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

      2. distribute-frac-neg 45.4%

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

   7. Simplified 45.4%

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

      1. associate-/r/ 41.9%

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

      2. add-sqr-sqrt 41.9%

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

      3. sqrt-unprod 52.3%

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

      4. sqr-neg 52.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 41.7%

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

   9. Applied egg-rr 41.7%

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

       1. frac-2neg 41.7%

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

       2. metadata-eval 41.7%

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

       3. associate-/r/ 45.3%

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

       4. add-sqr-sqrt 45.3%

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

       5. sqrt-unprod 53.5%

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

       6. sqr-neg 53.5%

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

       7. sqrt-unprod -0.0%

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

       8. add-sqr-sqrt 45.4%

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

   11. Applied egg-rr 45.4%

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

4. Final simplification 49.3%

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

Alternative 7: 45.7% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999747378752e-5) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999747378752e-5f) {
		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 <= (-9.999999747378752e-5)) 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(-9.999999747378752e-5))
		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(-9.999999747378752e-5))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999975e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. unsub-neg 56.9%

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

   4. Simplified 56.9%

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

   5. Taylor expanded in x around inf 56.9%

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

      1. mul-1-neg 56.9%

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

      2. distribute-frac-neg 56.9%

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

   7. Simplified 56.9%

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

      1. expm1-log1p-u 56.9%

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

      2. expm1-udef 97.6%

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

      3. clear-num 97.6%

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

      4. add-sqr-sqrt 97.6%

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

      5. sqrt-unprod 97.6%

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

      6. sqr-neg 97.6%

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

      7. sqrt-unprod -0.0%

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

      8. add-sqr-sqrt 97.6%

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

   9. Applied egg-rr 97.6%

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

       1. expm1-def 52.2%

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

       2. expm1-log1p 52.2%

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

   11. Simplified 52.2%

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

   if -9.99999975e-5 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 45.8%

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

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999747378752 \cdot 10^{-5}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 34.7% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 33.8%

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

3. Final simplification 33.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023322 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))