Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 15.1s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\frac{x}{s} \cdot -0.6666666666666666} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)} + 1} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   (* (exp (* (/ x s) -0.6666666666666666)) (pow (cbrt (exp -1.0)) (/ x s)))
   1.0)))

C

float code(float x, float s) {
	return 1.0f / ((expf(((x / s) * -0.6666666666666666f)) * powf(cbrtf(expf(-1.0f)), (x / s))) + 1.0f);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(exp(Float32(Float32(x / s) * Float32(-0.6666666666666666))) * (cbrt(exp(Float32(-1.0))) ^ Float32(x / s))) + Float32(1.0)))
end

Derivation

1. Initial program 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. distribute-frac-neg 99.9%

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

   4. neg-mul-1 99.9%

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

   5. add-sqr-sqrt 49.6%

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

   6. sqrt-unprod 60.3%

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

   7. sqr-neg 60.3%

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

   8. sqrt-unprod 12.5%

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

   9. add-sqr-sqrt 27.4%

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

   10. pow-unpow 27.4%

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

   11. add-sqr-sqrt 12.5%

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

   12. sqrt-unprod 60.3%

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

   13. sqr-neg 60.3%

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

   14. sqrt-unprod 49.6%

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

   15. add-sqr-sqrt 99.9%

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

4. Applied egg-rr 99.9%

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

   1. add-cube-cbrt 99.7%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}} \cdot \sqrt[3]{{\left(e^{1}\right)}^{-1}}\right) \cdot \sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}}^{\left(\frac{x}{s}\right)}} \]

   2. unpow-prod-down 99.8%

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}} \cdot \sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}}} \]

   3. cbrt-unprod 99.8%

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\sqrt[3]{{\left(e^{1}\right)}^{-1} \cdot {\left(e^{1}\right)}^{-1}}\right)}}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   4. pow-exp 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{\color{blue}{e^{1 \cdot -1}} \cdot {\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   5. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{\color{blue}{-1}} \cdot {\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   6. pow-exp 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{-1} \cdot \color{blue}{e^{1 \cdot -1}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   7. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{-1} \cdot e^{\color{blue}{-1}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   8. prod-exp 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{\color{blue}{e^{-1 + -1}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   9. metadata-eval 99.8%

      \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{\color{blue}{-2}}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{{\left(e^{1}\right)}^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   10. pow-exp 99.8%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{\color{blue}{e^{1 \cdot -1}}}\right)}^{\left(\frac{x}{s}\right)}} \]

   11. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + {\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{\color{blue}{-1}}}\right)}^{\left(\frac{x}{s}\right)}} \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}}} \]
7. Step-by-step derivation

   1. add-exp-log 99.9%

      \[\leadsto \frac{1}{1 + \color{blue}{e^{\log \left({\left(\sqrt[3]{e^{-2}}\right)}^{\left(\frac{x}{s}\right)}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   2. log-pow 99.8%

      \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{x}{s} \cdot \log \left(\sqrt[3]{e^{-2}}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   3. pow1/3 99.8%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \log \color{blue}{\left({\left(e^{-2}\right)}^{0.3333333333333333}\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   4. log-pow 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{-2}\right)\right)}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   5. rem-log-exp 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \left(0.3333333333333333 \cdot \color{blue}{-2}\right)} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

   6. metadata-eval 99.9%

      \[\leadsto \frac{1}{1 + e^{\frac{x}{s} \cdot \color{blue}{-0.6666666666666666}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

8. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{e^{\frac{x}{s} \cdot -0.6666666666666666}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

9. Final simplification 99.9%

   \[\leadsto \frac{1}{e^{\frac{x}{s} \cdot -0.6666666666666666} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)} + 1} \]
10. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. distribute-frac-neg 99.9%

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

   4. neg-mul-1 99.9%

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

   5. add-sqr-sqrt 49.6%

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

   6. sqrt-unprod 60.3%

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

   7. sqr-neg 60.3%

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

   8. sqrt-unprod 12.5%

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

   9. add-sqr-sqrt 27.4%

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

   10. pow-unpow 27.4%

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

   11. add-sqr-sqrt 12.5%

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

   12. sqrt-unprod 60.3%

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

   13. sqr-neg 60.3%

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

   14. sqrt-unprod 49.6%

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

   15. add-sqr-sqrt 99.9%

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

4. Applied egg-rr 99.9%

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

   1. add-sqr-sqrt 99.9%

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

   2. unpow-prod-down 99.8%

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

   3. sqrt-pow1 99.8%

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

   4. pow-to-exp 99.8%

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

   5. rem-log-exp 99.8%

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

   6. metadata-eval 99.8%

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

   7. metadata-eval 99.8%

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

   8. sqrt-pow1 99.8%

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

   9. pow-to-exp 99.8%

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

   10. rem-log-exp 99.8%

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

   11. metadata-eval 99.8%

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

   12. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

   1. pow-sqr 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-*l/ 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(((Float32(exp(1)) ^ Float32(-1.0)) ^ Float32(x / s)) + Float32(1.0)))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (((single(2.71828182845904523536) ^ single(-1.0)) ^ (x / s)) + single(1.0));
end

Derivation

1. Initial program 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.9%

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

   3. distribute-frac-neg 99.9%

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

   4. neg-mul-1 99.9%

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

   5. add-sqr-sqrt 49.6%

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

   6. sqrt-unprod 60.3%

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

   7. sqr-neg 60.3%

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

   8. sqrt-unprod 12.5%

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

   9. add-sqr-sqrt 27.4%

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

   10. pow-unpow 27.4%

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

   11. add-sqr-sqrt 12.5%

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

   12. sqrt-unprod 60.3%

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

   13. sqr-neg 60.3%

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

   14. sqrt-unprod 49.6%

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

   15. add-sqr-sqrt 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 4: 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 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 48.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(x / s) + Float32(2.0))));
	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(0.5);
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / ((x / s) + single(2.0)));
	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 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) < +inf.0

   1. Initial program 99.8%

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

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

      1. sub-neg 59.5%

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

      2. neg-mul-1 59.5%

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

      3. add-log-exp 96.9%

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

      4. pow-exp 96.9%

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

      5. metadata-eval 96.9%

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

      6. pow-exp 96.9%

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

      7. flip-+ 37.6%

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

   7. Applied egg-rr 56.4%

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

      1. distribute-frac-neg2 56.4%

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

      2. distribute-frac-neg2 56.4%

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

      3. sqr-neg 56.4%

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

      4. clear-num 56.4%

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

      5. frac-times 58.6%

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

      6. *-un-lft-identity 58.6%

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

   9. Applied egg-rr 58.6%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in x around inf 17.7%

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

      1. associate-*r/ 17.7%

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

      2. neg-mul-1 17.7%

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

   8. Simplified 17.7%

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

      1. inv-pow 17.7%

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

      2. distribute-frac-neg 17.7%

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

      3. distribute-frac-neg2 17.7%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 27.5%

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

      6. sqr-neg 27.5%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 20.0%

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

   10. Applied egg-rr 20.0%

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

       1. unpow-1 20.0%

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

   12. Simplified 20.0%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{-s}\\ \mathbf{if}\;t\_0 \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ x (- s))))
   (if (<= t_0 0.5)
     0.5
     (if (<= t_0 INFINITY)
       (/ -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 <= 0.5f) {
		tmp = 0.5f;
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = -1.0f / ((((x / s) * (x / s)) - 4.0f) / (x / s));
	} else {
		tmp = 1.0f / (x / s);
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(x / Float32(-s))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.5))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(Inf))
		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(0.5))
		tmp = single(0.5);
	elseif (t_0 <= single(Inf))
		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) < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.2%

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

   if 0.5 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. unsub-neg 36.2%

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

   5. Simplified 36.2%

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

      1. sub-neg 36.2%

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

      2. neg-mul-1 36.2%

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

      3. add-log-exp 97.5%

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

      4. pow-exp 97.5%

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

      5. metadata-eval 97.5%

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

      6. pow-exp 97.5%

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

      7. flip-+ 0.5%

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

   7. Applied egg-rr 31.1%

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

   8. Taylor expanded in x around inf 31.1%

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

   if +inf.0 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 40.6%

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

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

   5. Simplified 40.6%

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

   6. Taylor expanded in x around inf 17.7%

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

      1. associate-*r/ 17.7%

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

      2. neg-mul-1 17.7%

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

   8. Simplified 17.7%

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

      1. inv-pow 17.7%

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

      2. distribute-frac-neg 17.7%

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

      3. distribute-frac-neg2 17.7%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 27.5%

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

      6. sqr-neg 27.5%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 20.0%

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

   10. Applied egg-rr 20.0%

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

       1. unpow-1 20.0%

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

   12. Simplified 20.0%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq 0.5:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{x}{-s} \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{\frac{x}{s} \cdot \frac{x}{s} - 4}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.6% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ x (- s)) -1.0) 0.5 (/ 1.0 (+ -1.0 (- 3.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= -1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (-1.0f + (3.0f - (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 / ((-1.0e0) + (3.0e0 - (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(-1.0) + Float32(Float32(3.0) - 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(-1.0) + (single(3.0) - (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 99.8%

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

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

      1. expm1-log1p-u 59.4%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]

   7. Applied egg-rr 59.4%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 - \frac{x}{s}\right)\right)}} \]
   8. Step-by-step derivation

      1. expm1-undefine 59.4%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} - 1}} \]

      2. sub-neg 59.4%

         \[\leadsto \frac{1}{\color{blue}{e^{\mathsf{log1p}\left(2 - \frac{x}{s}\right)} + \left(-1\right)}} \]

      3. log1p-undefine 59.4%

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

      4. rem-exp-log 59.5%

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

      5. associate-+r- 59.5%

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

      6. metadata-eval 59.5%

         \[\leadsto \frac{1}{\left(\color{blue}{3} - \frac{x}{s}\right) + \left(-1\right)} \]

      7. metadata-eval 59.5%

         \[\leadsto \frac{1}{\left(3 - \frac{x}{s}\right) + \color{blue}{-1}} \]

   9. Simplified 59.5%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{-s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-1 + \left(3 - \frac{x}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.6% 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 99.8%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 48.6%

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

Alternative 9: 48.0% accurate, 8.3× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((x / -s) <= 0.5f) {
		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) <= 0.5e0) 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(0.5))
		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(0.5))
		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) < 0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 54.2%

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

   if 0.5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. unsub-neg 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 36.2%

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

      1. associate-*r/ 36.2%

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

      2. neg-mul-1 36.2%

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

   8. Simplified 36.2%

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

4. Final simplification 47.0%

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

Alternative 10: 46.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000116860974e-7) (* s (/ -1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000116860974e-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 <= (-1.0000000116860974e-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(-1.0000000116860974e-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(-1.0000000116860974e-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 < -1.00000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.7%

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

      1. mul-1-neg 44.7%

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

      2. unsub-neg 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. neg-mul-1 44.7%

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

   8. Simplified 44.7%

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

      1. associate-/r/ 40.8%

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

      2. frac-2neg 40.8%

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

      3. metadata-eval 40.8%

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

      4. remove-double-neg 40.8%

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

   10. Applied egg-rr 40.8%

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

   if -1.00000001e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.7% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.500000053056283e-7) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.500000053056283e-7f) {
		tmp = 1.0f / (x / s);
	} 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 <= (-1.500000053056283e-7)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.500000053056283e-7))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.500000053056283e-7))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.5000001e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. unsub-neg 45.1%

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

   5. Simplified 45.1%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. inv-pow 45.1%

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

      2. distribute-frac-neg 45.1%

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

      3. distribute-frac-neg2 45.1%

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

      4. add-sqr-sqrt -0.0%

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

      5. sqrt-unprod 51.8%

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

      6. sqr-neg 51.8%

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

      7. sqrt-unprod 45.1%

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

      8. add-sqr-sqrt 45.1%

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

   10. Applied egg-rr 45.1%

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

       1. unpow-1 45.1%

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

   12. Simplified 45.1%

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

   if -1.5000001e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.7%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000116860974e-7) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000116860974e-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 <= (-1.0000000116860974e-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(-1.0000000116860974e-7))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000116860974e-7))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.00000001e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 44.7%

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

      1. mul-1-neg 44.7%

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

      2. unsub-neg 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in x around inf 40.8%

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

      1. associate-*r/ 40.8%

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

      2. neg-mul-1 40.8%

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

   8. Simplified 40.8%

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

   if -1.00000001e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.9%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.4% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.500000053056283e-7) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.500000053056283e-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 <= (-1.500000053056283e-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(-1.500000053056283e-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(-1.500000053056283e-7))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.5000001e-7

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 45.1%

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

      1. mul-1-neg 45.1%

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

      2. unsub-neg 45.1%

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

   5. Simplified 45.1%

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

   6. Taylor expanded in x around inf 45.1%

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

      1. associate-*r/ 45.1%

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

      2. neg-mul-1 45.1%

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

   8. Simplified 45.1%

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

      1. clear-num 41.2%

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

      2. add-sqr-sqrt 41.2%

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

      3. sqrt-unprod 55.9%

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

      4. sqr-neg 55.9%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 41.2%

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

      7. *-un-lft-identity 41.2%

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

   10. Applied egg-rr 41.2%

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

       1. *-lft-identity 41.2%

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

   12. Simplified 41.2%

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

   if -1.5000001e-7 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 47.7%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.5% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0 35.2%

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

4. Final simplification 35.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))