Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 15.5s

Alternatives: 15

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. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. neg-mul-1 99.7%

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

   5. add-sqr-sqrt 47.2%

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

   6. sqrt-unprod 59.5%

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

   7. sqr-neg 59.5%

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

   8. sqrt-unprod 14.5%

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

   9. add-sqr-sqrt 29.4%

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

   10. pow-unpow 29.4%

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

   11. add-sqr-sqrt 14.5%

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

   12. sqrt-unprod 59.5%

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

   13. sqr-neg 59.5%

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

   14. sqrt-unprod 47.2%

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

   15. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-exp-log 99.7%

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

   2. log-rec 99.7%

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

   3. log1p-define 99.8%

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

   4. pow-exp 99.8%

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

   5. metadata-eval 99.8%

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

   6. pow-exp 99.8%

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

   7. add-log-exp 99.7%

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

   8. log-pow 99.7%

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

   9. inv-pow 99.7%

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

   10. neg-log 99.7%

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

   11. add-log-exp 99.8%

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

   12. distribute-neg-frac2 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x \cdot 2}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp -0.5) (/ (* x 2.0) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-0.5f), ((x * 2.0f) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((-0.5e0)) ** ((x * 2.0e0) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-0.5)) ^ Float32(Float32(x * Float32(2.0)) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-0.5)) ^ ((x * single(2.0)) / s)));
end

Derivation

1. Initial program 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. exp-prod 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. neg-mul-1 99.7%

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

   5. add-sqr-sqrt 47.2%

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

   6. sqrt-unprod 59.5%

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

   7. sqr-neg 59.5%

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

   8. sqrt-unprod 14.5%

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

   9. add-sqr-sqrt 29.4%

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

   10. pow-unpow 29.4%

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

   11. add-sqr-sqrt 14.5%

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

   12. sqrt-unprod 59.5%

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

   13. sqr-neg 59.5%

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

   14. sqrt-unprod 47.2%

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

   15. add-sqr-sqrt 99.8%

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

4. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. unpow-prod-down 99.7%

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

   3. sqrt-pow1 99.7%

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

   4. pow-to-exp 99.7%

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

   5. rem-log-exp 99.7%

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

   6. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + {\left(e^{1 \cdot \color{blue}{-0.5}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   7. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + {\left(e^{\color{blue}{-0.5}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   8. sqrt-pow1 99.7%

      \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\color{blue}{\left({\left(e^{1}\right)}^{\left(\frac{-1}{2}\right)}\right)}}^{\left(\frac{x}{s}\right)}} \]

   9. pow-to-exp 99.7%

      \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\color{blue}{\left(e^{\log \left(e^{1}\right) \cdot \frac{-1}{2}}\right)}}^{\left(\frac{x}{s}\right)}} \]

   10. rem-log-exp 99.7%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{1} \cdot \frac{-1}{2}}\right)}^{\left(\frac{x}{s}\right)}} \]

   11. metadata-eval 99.7%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{1 \cdot \color{blue}{-0.5}}\right)}^{\left(\frac{x}{s}\right)}} \]

   12. metadata-eval 99.7%

       \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{\color{blue}{-0.5}}\right)}^{\left(\frac{x}{s}\right)}} \]

6. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(e^{-0.5}\right)}^{\left(\frac{x}{s}\right)}}} \]
7. Step-by-step derivation

   1. pow-sqr 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.5}\right)}^{\left(2 \cdot \frac{x}{s}\right)}}} \]

   2. *-commutative 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\color{blue}{\left(\frac{x}{s} \cdot 2\right)}}} \]

   3. associate-*l/ 99.8%

      \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\color{blue}{\left(\frac{x \cdot 2}{s}\right)}}} \]

8. Simplified 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{-0.5}\right)}^{\left(\frac{x \cdot 2}{s}\right)}}} \]

9. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + {\left(e^{-0.5}\right)}^{\left(\frac{x \cdot 2}{s}\right)}} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-1.0f), (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((-1.0e0)) ** (x / s)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. exp-neg 99.7%

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

4. Applied egg-rr 99.7%

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

   1. inv-pow 99.7%

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

   2. pow-exp 99.7%

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

   3. *-commutative 99.7%

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

   4. pow-exp 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + {\left(e^{-1}\right)}^{\left(\frac{x}{s}\right)}} \]
8. Add Preprocessing

Alternative 4: 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(x / Float32(-s)))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 5: 85.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 + \frac{-1}{\frac{s}{x} \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (+ 4.0 (/ -1.0 (* (/ s x) (/ s x)))) (/ x s)))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f + (-1.0f / ((s / x) * (s / x)))) / (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 <= 1.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 + ((-1.0e0) / ((s / x) * (s / x)))) / (x / s))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) + Float32(Float32(-1.0) / Float32(Float32(s / x) * Float32(s / x)))) / 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(1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) + (single(-1.0) / ((s / x) * (s / x)))) / (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) < 1

   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 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 96.1%

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

   if 1 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unsub-neg 10.7%

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

   5. Simplified 10.7%

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

      1. sub-neg 10.7%

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

      2. flip-+ 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-neg-frac2 42.9%

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

      5. distribute-neg-frac2 42.9%

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

      6. distribute-neg-frac2 42.9%

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

   7. Applied egg-rr 42.9%

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

      1. distribute-frac-neg2 42.9%

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

      2. distribute-frac-neg2 42.9%

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

      3. sqr-neg 42.9%

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

      4. clear-num 42.9%

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

      5. clear-num 42.9%

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

      6. frac-times 42.9%

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

      7. metadata-eval 42.9%

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

   9. Applied egg-rr 42.9%

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

   10. Taylor expanded in x around inf 42.9%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 + \frac{-1}{\frac{s}{x} \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -0.019999999552965164)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ 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 <= -0.019999999552965164f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		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 <= (-0.019999999552965164e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / (2.0e0 + (x / s)))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.019999999552965164))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = x / -s;
	tmp = single(0.0);
	if (t_0 <= single(-0.019999999552965164))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		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) < -0.0199999996

   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 97.1%

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

   if -0.0199999996 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. unsub-neg 52.2%

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

   5. Simplified 52.2%

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

      1. sub-neg 52.2%

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

      2. flip-+ 68.6%

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

      3. metadata-eval 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. distribute-neg-frac2 68.6%

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

      6. distribute-neg-frac2 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. distribute-frac-neg2 68.6%

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

      2. distribute-frac-neg2 68.6%

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

      3. sqr-neg 68.6%

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

      4. clear-num 68.6%

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

      5. un-div-inv 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. div-inv 68.6%

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

       2. clear-num 68.6%

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

   11. Applied egg-rr 68.6%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 -0.019999999552965164)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) 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 <= -0.019999999552965164f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / 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 <= (-0.019999999552965164e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / (2.0e0 + (x / s)))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-0.019999999552965164))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / 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(-0.019999999552965164))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / 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) < -0.0199999996

   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 97.1%

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

   if -0.0199999996 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 52.2%

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

      1. mul-1-neg 52.2%

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

      2. unsub-neg 52.2%

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

   5. Simplified 52.2%

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

      1. sub-neg 52.2%

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

      2. flip-+ 68.6%

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

      3. metadata-eval 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. distribute-neg-frac2 68.6%

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

      6. distribute-neg-frac2 68.6%

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

   7. Applied egg-rr 68.6%

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

      1. distribute-frac-neg2 68.6%

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

      2. distribute-frac-neg2 68.6%

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

      3. sqr-neg 68.6%

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

      4. clear-num 68.6%

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

      5. un-div-inv 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-/r/ 74.3%

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

   11. Applied egg-rr 74.3%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{\frac{x}{s}}{\frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ (/ x s) (/ s x))) (/ x s)))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - ((x / s) / (s / x))) / (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 <= 1.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) / (s / x))) / (x / s))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) / Float32(s / x))) / 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(1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - ((x / s) / (s / x))) / (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) < 1

   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 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 96.1%

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

   if 1 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unsub-neg 10.7%

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

   5. Simplified 10.7%

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

      1. sub-neg 10.7%

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

      2. flip-+ 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-neg-frac2 42.9%

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

      5. distribute-neg-frac2 42.9%

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

      6. distribute-neg-frac2 42.9%

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

   7. Applied egg-rr 42.9%

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

      1. distribute-frac-neg2 42.9%

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

      2. distribute-frac-neg2 42.9%

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

      3. sqr-neg 42.9%

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

      4. clear-num 42.9%

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

      5. un-div-inv 42.9%

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

   9. Applied egg-rr 42.9%

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

   10. Taylor expanded in x around inf 42.9%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{\frac{x}{s}}{\frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t\_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 1.0)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 1.9999999360571385e+38)
       (/ -1.0 (/ (- (* (/ x s) (/ x s)) 4.0) (/ x s)))
       (/ -1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = x / -s;
	float tmp;
	if (t_0 <= 1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (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 <= 1.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = (-1.0e0) / ((((x / s) * (x / s)) - 4.0e0) / (x / s))
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(-1.0) / Float32(Float32(Float32(Float32(x / s) * Float32(x / s)) - Float32(4.0)) / Float32(x / s)));
	else
		tmp = Float32(Float32(-1.0) / Float32(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(1.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(-1.0) / ((((x / s) * (x / s)) - single(4.0)) / (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) < 1

   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 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 96.1%

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

   if 1 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. unsub-neg 10.7%

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

   5. Simplified 10.7%

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

      1. sub-neg 10.7%

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

      2. flip-+ 42.9%

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

      3. metadata-eval 42.9%

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

      4. distribute-neg-frac2 42.9%

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

      5. distribute-neg-frac2 42.9%

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

      6. distribute-neg-frac2 42.9%

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

   7. Applied egg-rr 42.9%

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

   8. Taylor expanded in x around inf 42.9%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg2 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 1:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{x}{-s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.00015999999595806003:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -0.00015999999595806003)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -0.00015999999595806003f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} 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) <= (-0.00015999999595806003e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(-0.00015999999595806003))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	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(-0.00015999999595806003))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	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) < -1.59999996e-4

   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 96.9%

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

   if -1.59999996e-4 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 59.5%

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

      1. mul-1-neg 59.5%

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

      2. unsub-neg 59.5%

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

   5. Simplified 59.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -0.00015999999595806003:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.4% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		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) <= (-1.0e0)) 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(x / Float32(-s)) <= Float32(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.2%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 59.8%

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

      1. mul-1-neg 59.8%

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

      2. unsub-neg 59.8%

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

   5. Simplified 59.8%

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

4. Final simplification 49.3%

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

Alternative 12: 47.8% accurate, 8.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 1.0f) {
		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) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(x / Float32(-s)) <= Float32(1.0))
		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(1.0))
		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) < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 55.8%

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

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

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 35.0%

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

      1. mul-1-neg 35.0%

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

      2. unsub-neg 35.0%

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

   5. Simplified 35.0%

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

   6. Taylor expanded in x around inf 35.0%

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

      1. mul-1-neg 35.0%

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

      2. distribute-frac-neg2 35.0%

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

   8. Simplified 35.0%

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

4. Final simplification 47.8%

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

Alternative 13: 46.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2000000424450263 \cdot 10^{-5}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.2000000424450263e-5) (- (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.2000000424450263e-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 <= (-1.2000000424450263e-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(-1.2000000424450263e-5))
		tmp = Float32(-Float32(s / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.2000000424450263e-5))
		tmp = -(s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.20000004e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. unsub-neg 48.4%

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

   5. Simplified 48.4%

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

   6. Taylor expanded in x around inf 42.8%

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

      1. associate-*r/ 42.8%

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

      2. neg-mul-1 42.8%

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

   8. Simplified 42.8%

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

   if -1.20000004e-5 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 47.4%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2000000424450263 \cdot 10^{-5}:\\ \;\;\;\;-\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.2% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.004999999888241291) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.004999999888241291f) {
		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 <= (-0.004999999888241291e0)) 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(-0.004999999888241291))
		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(-0.004999999888241291))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00499999989

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 54.1%

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

      1. mul-1-neg 54.1%

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

      2. unsub-neg 54.1%

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

   5. Simplified 54.1%

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

      1. sub-neg 54.1%

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

      2. flip-+ 39.8%

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

      3. metadata-eval 39.8%

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

      4. distribute-neg-frac2 39.8%

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

      5. distribute-neg-frac2 39.8%

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

      6. distribute-neg-frac2 39.8%

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

   7. Applied egg-rr 39.8%

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

   8. Taylor expanded in x around inf 54.1%

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

      1. associate-*r/ 54.1%

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

      2. neg-mul-1 54.1%

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

   10. Simplified 54.1%

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

       1. add-sqr-sqrt 54.1%

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

       2. sqrt-unprod 63.9%

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

       3. sqr-neg 63.9%

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

       4. sqrt-unprod -0.0%

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

       5. add-sqr-sqrt 54.1%

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

       6. clear-num 48.4%

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

       7. div-inv 48.4%

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

   12. Applied egg-rr 48.4%

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

       1. associate-*r/ 48.4%

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

       2. *-rgt-identity 48.4%

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

   14. Simplified 48.4%

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

   if -0.00499999989 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 45.6%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.004999999888241291:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.8% 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 36.8%

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

4. Final simplification 36.8%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024062 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))