Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 8.8s

Alternatives: 12

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(x / Float32(-s))))))
end

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 83.1%

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

   3. neg-mul-1 83.1%

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

   4. exp-prod 83.1%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left(\mathsf{expm1}\left(\frac{x}{-s}\right) + 1\right) + 1} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ (+ (expm1 (/ x (- s))) 1.0) 1.0)))

C

float code(float x, float s) {
	return 1.0f / ((expm1f((x / -s)) + 1.0f) + 1.0f);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(expm1(Float32(x / Float32(-s))) + Float32(1.0)) + Float32(1.0)))
end

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.7%

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

   2. exp-prod 83.1%

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

   3. neg-mul-1 83.1%

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

   4. exp-prod 83.1%

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

   5. pow-pow 99.7%

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

   6. div-inv 99.7%

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

4. Applied egg-rr 99.7%

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

   1. pow-exp 99.7%

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

   2. *-commutative 99.7%

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

   3. pow-exp 99.7%

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

   4. inv-pow 99.7%

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

   5. expm1-log1p-u 99.7%

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

   6. log1p-define 99.7%

      \[\leadsto \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{1}{e^{\frac{x}{s}}}\right)}\right)} \]

   7. expm1-undefine 99.7%

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

   8. add-exp-log 99.7%

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

   9. rec-exp 99.7%

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

   10. distribute-neg-frac2 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate--l+ 99.7%

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

   2. expm1-define 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{1}{\left(\mathsf{expm1}\left(\frac{x}{-s}\right) + 1\right) + 1} \]
10. Add Preprocessing

Alternative 3: 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 99.7%

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

3. Final simplification 99.7%

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

Alternative 4: 76.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000100204387e+20)
   (/ 1.0 (/ x s))
   (if (<= x 9.999999998199587e-24)
     (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
     1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000100204387e+20f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-5.000000100204387e+20)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 9.999999998199587e-24) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.000000100204387e+20))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-5.000000100204387e+20))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -5.0000001e20

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 91.1%

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

      1. neg-mul-1 91.1%

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

      2. unsub-neg 91.1%

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

   5. Simplified 91.1%

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

   6. Taylor expanded in x around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. neg-mul-1 84.3%

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

   8. Simplified 84.3%

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

      1. clear-num 91.1%

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

      2. inv-pow 91.1%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 91.1%

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

      5. sqr-neg 91.1%

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

      6. sqrt-unprod 91.1%

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

      7. add-sqr-sqrt 91.1%

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

   10. Applied egg-rr 91.1%

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

       1. unpow-1 91.1%

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

   12. Simplified 91.1%

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

   if -5.0000001e20 < x < 1e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 58.0%

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

      1. neg-mul-1 58.0%

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

      2. unsub-neg 58.0%

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

   5. Simplified 58.0%

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

      1. *-un-lft-identity 58.0%

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

      2. cancel-sign-sub-inv 58.0%

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

      3. metadata-eval 58.0%

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

      4. add-log-exp 91.9%

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

      5. pow-exp 91.9%

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

      6. flip-+ 50.0%

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

      7. metadata-eval 50.0%

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

      8. pow-exp 50.0%

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

      9. add-log-exp 50.0%

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

      10. neg-mul-1 50.0%

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

      11. pow-exp 50.0%

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

      12. add-log-exp 50.6%

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

      13. neg-mul-1 50.6%

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

      14. distribute-neg-frac2 50.6%

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

      15. distribute-neg-frac2 50.6%

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

      16. pow-exp 50.6%

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

   7. Applied egg-rr 71.6%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 94.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.000000100204387 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999974752427e-7)
   (/ 1.0 (/ x s))
   (if (<= x 9.999999998199587e-24) (+ 0.5 (/ (* x 0.25) s)) 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-7f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999974752427e-7)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999974752427e-7))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999974752427e-7))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. clear-num 48.7%

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

      2. inv-pow 48.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 48.7%

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

      7. add-sqr-sqrt 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. unpow-1 48.7%

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

   12. Simplified 48.7%

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

   if -9.99999997e-7 < x < 1e-23

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 74.6%

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

      1. associate-*r/ 74.6%

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

   5. Simplified 74.6%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 94.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999974752427e-7)
   (/ 1.0 (/ x s))
   (if (<= x 9.999999998199587e-24) (+ 0.5 (/ (* x 0.25) s)) (+ (/ s x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-7f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = (s / x) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999974752427e-7)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = (s / x) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999974752427e-7))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(s / x) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999974752427e-7))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = (s / x) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. clear-num 48.7%

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

      2. inv-pow 48.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 48.7%

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

      7. add-sqr-sqrt 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. unpow-1 48.7%

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

   12. Simplified 48.7%

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

   if -9.99999997e-7 < x < 1e-23

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 74.6%

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

      1. associate-*r/ 74.6%

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

   5. Simplified 74.6%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 90.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999974752427e-7)
   (/ 1.0 (/ x s))
   (if (<= x 9.999999998199587e-24) 0.5 (+ (/ s x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-7f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = (s / x) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999974752427e-7)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = (s / x) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999974752427e-7))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s / x) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999974752427e-7))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = (s / x) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. clear-num 48.7%

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

      2. inv-pow 48.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 63.9%

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

      5. sqr-neg 63.9%

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

      6. sqrt-unprod 48.7%

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

      7. add-sqr-sqrt 48.7%

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

   10. Applied egg-rr 48.7%

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

       1. unpow-1 48.7%

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

   12. Simplified 48.7%

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

   if -9.99999997e-7 < x < 1e-23

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 71.0%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 90.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999974752427e-7)
   (* s (/ 1.0 x))
   (if (<= x 9.999999998199587e-24) 0.5 (+ (/ s x) 1.0))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-7f) {
		tmp = s * (1.0f / x);
	} else if (x <= 9.999999998199587e-24f) {
		tmp = 0.5f;
	} else {
		tmp = (s / x) + 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999974752427e-7)) then
        tmp = s * (1.0e0 / x)
    else if (x <= 9.999999998199587e-24) then
        tmp = 0.5e0
    else
        tmp = (s / x) + 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999974752427e-7))
		tmp = Float32(s * Float32(Float32(1.0) / x));
	elseif (x <= Float32(9.999999998199587e-24))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s / x) + Float32(1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999974752427e-7))
		tmp = s * (single(1.0) / x);
	elseif (x <= single(9.999999998199587e-24))
		tmp = single(0.5);
	else
		tmp = (s / x) + single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. clear-num 48.7%

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

      2. associate-/r/ 44.5%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 44.5%

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

      7. add-sqr-sqrt 44.5%

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

   10. Applied egg-rr 44.5%

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

   if -9.99999997e-7 < x < 1e-23

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 71.0%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 90.3%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999974752427 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s}{x} + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 9.999999998199587e-24) (/ 1.0 (- 2.0 (/ x s))) 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 9.999999998199587e-24f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 9.999999998199587e-24) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(9.999999998199587e-24))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(9.999999998199587e-24))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 63.2%

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

      1. neg-mul-1 63.2%

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

      2. unsub-neg 63.2%

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

   5. Simplified 63.2%

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

   if 1e-23 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. +-commutative 92.1%

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

   7. Simplified 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. log1p-define 92.1%

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

      3. expm1-undefine 92.1%

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

   9. Applied egg-rr 91.4%

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

       1. associate--l+ 91.4%

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

       2. sub-neg 91.4%

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

       3. +-commutative 91.4%

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

       4. metadata-eval 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around inf 94.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.999999998199587 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.0% accurate, 10.8× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999974752427e-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 <= (-9.999999974752427e-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(-9.999999974752427e-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(-9.999999974752427e-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 < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. clear-num 48.7%

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

      2. associate-/r/ 44.5%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 44.5%

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

      7. add-sqr-sqrt 44.5%

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

   10. Applied egg-rr 44.5%

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

   if -9.99999997e-7 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 49.8%

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

4. Final simplification 48.4%

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

Alternative 11: 46.0% accurate, 13.4× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if x < -9.99999997e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.7%

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

      1. neg-mul-1 48.7%

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

      2. unsub-neg 48.7%

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

   5. Simplified 48.7%

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

   6. Taylor expanded in x around inf 44.5%

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

      1. associate-*r/ 44.5%

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

      2. neg-mul-1 44.5%

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

   8. Simplified 44.5%

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

      1. neg-sub0 44.5%

         \[\leadsto \frac{\color{blue}{0 - s}}{x} \]

      2. sub-neg 44.5%

         \[\leadsto \frac{\color{blue}{0 + \left(-s\right)}}{x} \]

      3. add-sqr-sqrt -0.0%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{x} \]

      4. sqrt-unprod 61.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x} \]

      5. sqr-neg 61.5%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{s \cdot s}}}{x} \]

      6. sqrt-unprod 44.5%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x} \]

      7. add-sqr-sqrt 44.5%

         \[\leadsto \frac{0 + \color{blue}{s}}{x} \]

   10. Applied egg-rr 44.5%

       \[\leadsto \frac{\color{blue}{0 + s}}{x} \]
   11. Step-by-step derivation

       1. +-lft-identity 44.5%

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

   12. Simplified 44.5%

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

   if -9.99999997e-7 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 49.8%

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

Alternative 12: 35.0% 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 99.7%

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

3. Taylor expanded in x around 0 38.6%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))