Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

Alternatives: 17

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

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{1}{\frac{s}{x} \cdot -1.5}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+ 1.0 (* (exp (/ 1.0 (* (/ s x) -1.5))) (pow (cbrt (exp -1.0)) (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf((1.0f / ((s / x) * -1.5f))) * powf(cbrtf(expf(-1.0f)), (x / s))));
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(exp(Float32(Float32(1.0) / Float32(Float32(s / x) * Float32(-1.5)))) * (cbrt(exp(Float32(-1.0))) ^ Float32(x / s)))))
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 45.7%

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

   4. sqrt-unprod 54.1%

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

   5. sqr-neg 54.1%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 22.7%

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

   8. add-cube-cbrt 22.7%

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

   9. pow3 22.7%

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

   10. pow-flip 22.7%

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

   11. add-sqr-sqrt 9.9%

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

   12. sqrt-unprod 54.1%

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

   13. sqr-neg 54.1%

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

   14. sqrt-unprod 45.7%

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

   15. add-sqr-sqrt 99.7%

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

   16. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. pow1/3 99.7%

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

   2. pow-pow 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-exp 99.8%

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

   5. *-commutative 99.8%

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

   6. pow-exp 99.7%

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

   7. add-cube-cbrt 99.6%

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

   8. unpow-prod-down 99.6%

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

   9. cbrt-unprod 99.7%

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

   10. prod-exp 99.7%

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

   11. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

   1. pow1/3 99.7%

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

   2. pow-exp 99.7%

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

   3. metadata-eval 99.7%

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

   4. exp-prod 99.8%

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

   5. associate-*r/ 99.8%

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

7. Applied egg-rr 99.8%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

9. Applied egg-rr 99.7%

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

    1. unpow-1 99.7%

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

    2. *-rgt-identity 99.7%

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

    3. *-commutative 99.7%

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

    4. times-frac 99.8%

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

    5. metadata-eval 99.8%

       \[\leadsto \frac{1}{1 + e^{\frac{1}{\frac{s}{x} \cdot \color{blue}{-1.5}}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

11. Simplified 99.8%

    \[\leadsto \frac{1}{1 + e^{\color{blue}{\frac{1}{\frac{s}{x} \cdot -1.5}}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

12. Final simplification 99.8%

    \[\leadsto \frac{1}{1 + e^{\frac{1}{\frac{s}{x} \cdot -1.5}} \cdot {\left(\sqrt[3]{e^{-1}}\right)}^{\left(\frac{x}{s}\right)}} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{x \cdot -0.6666666666666666}{s}} \cdot e^{\frac{x}{s} \cdot -0.3333333333333333}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (+
   1.0
   (*
    (exp (/ (* x -0.6666666666666666) s))
    (exp (* (/ x s) -0.3333333333333333))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + (expf(((x * -0.6666666666666666f) / s)) * expf(((x / s) * -0.3333333333333333f))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp(((x * (-0.6666666666666666e0)) / s)) * exp(((x / s) * (-0.3333333333333333e0)))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(exp(Float32(Float32(x * Float32(-0.6666666666666666)) / s)) * exp(Float32(Float32(x / s) * Float32(-0.3333333333333333))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(((x * single(-0.6666666666666666)) / s)) * exp(((x / s) * single(-0.3333333333333333)))));
end

Derivation

1. Initial program 99.8%

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

   1. distribute-frac-neg 99.8%

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

   2. exp-neg 99.7%

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

   3. add-sqr-sqrt 45.7%

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

   4. sqrt-unprod 54.1%

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

   5. sqr-neg 54.1%

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

   6. sqrt-unprod 9.9%

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

   7. add-sqr-sqrt 22.7%

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

   8. add-cube-cbrt 22.7%

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

   9. pow3 22.7%

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

   10. pow-flip 22.7%

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

   11. add-sqr-sqrt 9.9%

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

   12. sqrt-unprod 54.1%

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

   13. sqr-neg 54.1%

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

   14. sqrt-unprod 45.7%

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

   15. add-sqr-sqrt 99.7%

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

   16. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. pow1/3 99.7%

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

   2. pow-pow 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-exp 99.8%

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

   5. *-commutative 99.8%

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

   6. pow-exp 99.7%

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

   7. add-cube-cbrt 99.6%

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

   8. unpow-prod-down 99.6%

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

   9. cbrt-unprod 99.7%

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

   10. prod-exp 99.7%

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

   11. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

   1. pow1/3 99.7%

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

   2. pow-exp 99.7%

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

   3. metadata-eval 99.7%

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

   4. exp-prod 99.8%

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

   5. associate-*r/ 99.8%

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

7. Applied egg-rr 99.8%

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

   1. pow1/3 99.8%

      \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot {\color{blue}{\left({\left(e^{-1}\right)}^{0.3333333333333333}\right)}}^{\left(\frac{x}{s}\right)}} \]

   2. pow-exp 99.7%

      \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot {\color{blue}{\left(e^{-1 \cdot 0.3333333333333333}\right)}}^{\left(\frac{x}{s}\right)}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot {\left(e^{\color{blue}{-0.3333333333333333}}\right)}^{\left(\frac{x}{s}\right)}} \]

   4. exp-prod 99.8%

      \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot \color{blue}{e^{-0.3333333333333333 \cdot \frac{x}{s}}}} \]

   5. *-commutative 99.8%

      \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot e^{\color{blue}{\frac{x}{s} \cdot -0.3333333333333333}}} \]

9. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + e^{\frac{-0.6666666666666666 \cdot x}{s}} \cdot \color{blue}{e^{\frac{x}{s} \cdot -0.3333333333333333}}} \]

10. Final simplification 99.8%

    \[\leadsto \frac{1}{1 + e^{\frac{x \cdot -0.6666666666666666}{s}} \cdot e^{\frac{x}{s} \cdot -0.3333333333333333}} \]

Alternative 3: 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

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 4: 88.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{s}{x \cdot -0.3333333333333333}\\ t_1 := \frac{-x}{s}\\ \mathbf{if}\;t_1 \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_1 \leq 200000000753664:\\ \;\;\;\;\frac{1}{1 + \left(1 - x \cdot \frac{s \cdot -0.3333333333333333 - s \cdot 0.6666666666666666}{s \cdot \left(-s\right)}\right)}\\ \mathbf{elif}\;t_1 \leq 9.999999616903162 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \frac{s + \left(x \cdot -0.6666666666666666\right) \cdot t_0}{s \cdot t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ s (* x -0.3333333333333333))) (t_1 (/ (- x) s)))
   (if (<= t_1 0.20000000298023224)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_1 200000000753664.0)
       (/
        1.0
        (+
         1.0
         (-
          1.0
          (*
           x
           (/
            (- (* s -0.3333333333333333) (* s 0.6666666666666666))
            (* s (- s)))))))
       (if (<= t_1 9.999999616903162e+35)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/
          1.0
          (+
           1.0
           (+ 1.0 (/ (+ s (* (* x -0.6666666666666666) t_0)) (* s t_0))))))))))

C

float code(float x, float s) {
	float t_0 = s / (x * -0.3333333333333333f);
	float t_1 = -x / s;
	float tmp;
	if (t_1 <= 0.20000000298023224f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_1 <= 200000000753664.0f) {
		tmp = 1.0f / (1.0f + (1.0f - (x * (((s * -0.3333333333333333f) - (s * 0.6666666666666666f)) / (s * -s)))));
	} else if (t_1 <= 9.999999616903162e+35f) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + ((s + ((x * -0.6666666666666666f) * t_0)) / (s * t_0))));
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = s / (x * (-0.3333333333333333e0))
    t_1 = -x / s
    if (t_1 <= 0.20000000298023224e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else if (t_1 <= 200000000753664.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 - (x * (((s * (-0.3333333333333333e0)) - (s * 0.6666666666666666e0)) / (s * -s)))))
    else if (t_1 <= 9.999999616903162e+35) then
        tmp = 1.0e0 / ((4.0e0 - ((x / s) * (x / s))) / ((x / s) + 2.0e0))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + ((s + ((x * (-0.6666666666666666e0)) * t_0)) / (s * t_0))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(s / Float32(x * Float32(-0.3333333333333333)))
	t_1 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_1 <= Float32(200000000753664.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(x * Float32(Float32(Float32(s * Float32(-0.3333333333333333)) - Float32(s * Float32(0.6666666666666666))) / Float32(s * Float32(-s)))))));
	elseif (t_1 <= Float32(9.999999616903162e+35))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(s + Float32(Float32(x * Float32(-0.6666666666666666)) * t_0)) / Float32(s * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = s / (x * single(-0.3333333333333333));
	t_1 = -x / s;
	tmp = single(0.0);
	if (t_1 <= single(0.20000000298023224))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_1 <= single(200000000753664.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) - (x * (((s * single(-0.3333333333333333)) - (s * single(0.6666666666666666))) / (s * -s)))));
	elseif (t_1 <= single(9.999999616903162e+35))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + ((s + ((x * single(-0.6666666666666666)) * t_0)) / (s * t_0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 97.2%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s) < 2.00000001e14

   1. Initial program 98.7%

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

      1. distribute-frac-neg 98.7%

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

      2. exp-neg 98.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 7.2%

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

      5. sqr-neg 7.2%

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

      6. sqrt-unprod 7.2%

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

      7. add-sqr-sqrt 7.2%

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

      8. add-cube-cbrt 7.2%

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

      9. pow3 7.2%

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

      10. pow-flip 7.2%

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

      11. add-sqr-sqrt 7.2%

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

      12. sqrt-unprod 7.2%

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

      13. sqr-neg 7.2%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 98.8%

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

      16. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in s around inf 8.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 8.3%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 8.3%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]
   9. Step-by-step derivation

      1. frac-2neg 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \left(\color{blue}{\frac{--0.6666666666666666}{-s}} + \frac{-0.3333333333333333}{s}\right)\right)} \]

      2. frac-add 42.2%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\frac{\left(--0.6666666666666666\right) \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}}\right)} \]

      3. metadata-eval 42.2%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{\color{blue}{0.6666666666666666} \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}\right)} \]

   10. Applied egg-rr 42.2%

       \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\frac{0.6666666666666666 \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}}\right)} \]

   if 2.00000001e14 < (/.f32 (neg.f32 x) s) < 9.99999962e35

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 12.5%

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

      1. mul-1-neg 12.5%

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

      2. unsub-neg 12.5%

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

   4. Simplified 12.5%

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

      1. sub-neg 12.5%

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

      2. flip-+ 88.0%

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

      3. metadata-eval 88.0%

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

      4. distribute-neg-frac 88.0%

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

      5. distribute-neg-frac 88.0%

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

      6. distribute-neg-frac 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 9.99999962e35 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 6.3%

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

      5. sqr-neg 6.3%

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

      6. sqrt-unprod 6.3%

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

      7. add-sqr-sqrt 6.3%

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

      8. add-cube-cbrt 6.3%

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

      9. pow3 6.3%

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

      10. pow-flip 6.3%

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

      11. add-sqr-sqrt 6.3%

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

      12. sqrt-unprod 6.3%

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

      13. sqr-neg 6.3%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 100.0%

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

      16. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in s around inf 92.7%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. associate-*r/ 92.7%

         \[\leadsto \frac{1}{1 + \left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + \color{blue}{\frac{-0.3333333333333333 \cdot x}{s}}\right)\right)} \]

      2. clear-num 92.7%

         \[\leadsto \frac{1}{1 + \left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + \color{blue}{\frac{1}{\frac{s}{-0.3333333333333333 \cdot x}}}\right)\right)} \]

   6. Applied egg-rr 92.7%

      \[\leadsto \frac{1}{1 + \left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + \color{blue}{\frac{1}{\frac{s}{-0.3333333333333333 \cdot x}}}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-/r* 92.7%

         \[\leadsto \frac{1}{1 + \left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + \frac{1}{\color{blue}{\frac{\frac{s}{-0.3333333333333333}}{x}}}\right)\right)} \]

   8. Simplified 92.7%

      \[\leadsto \frac{1}{1 + \left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + \color{blue}{\frac{1}{\frac{\frac{s}{-0.3333333333333333}}{x}}}\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative 92.7%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(\frac{1}{\frac{\frac{s}{-0.3333333333333333}}{x}} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. associate-*r/ 92.7%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\frac{1}{\frac{\frac{s}{-0.3333333333333333}}{x}} + \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

      3. frac-add 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\frac{1 \cdot s + \frac{\frac{s}{-0.3333333333333333}}{x} \cdot \left(-0.6666666666666666 \cdot x\right)}{\frac{\frac{s}{-0.3333333333333333}}{x} \cdot s}}\right)} \]

      4. *-un-lft-identity 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + \frac{\color{blue}{s} + \frac{\frac{s}{-0.3333333333333333}}{x} \cdot \left(-0.6666666666666666 \cdot x\right)}{\frac{\frac{s}{-0.3333333333333333}}{x} \cdot s}\right)} \]

      5. associate-/l/ 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + \frac{s + \color{blue}{\frac{s}{x \cdot -0.3333333333333333}} \cdot \left(-0.6666666666666666 \cdot x\right)}{\frac{\frac{s}{-0.3333333333333333}}{x} \cdot s}\right)} \]

      6. associate-/l/ 100.0%

         \[\leadsto \frac{1}{1 + \left(1 + \frac{s + \frac{s}{x \cdot -0.3333333333333333} \cdot \left(-0.6666666666666666 \cdot x\right)}{\color{blue}{\frac{s}{x \cdot -0.3333333333333333}} \cdot s}\right)} \]

   10. Applied egg-rr 100.0%

       \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\frac{s + \frac{s}{x \cdot -0.3333333333333333} \cdot \left(-0.6666666666666666 \cdot x\right)}{\frac{s}{x \cdot -0.3333333333333333} \cdot s}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 200000000753664:\\ \;\;\;\;\frac{1}{1 + \left(1 - x \cdot \frac{s \cdot -0.3333333333333333 - s \cdot 0.6666666666666666}{s \cdot \left(-s\right)}\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999616903162 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + \frac{s + \left(x \cdot -0.6666666666666666\right) \cdot \frac{s}{x \cdot -0.3333333333333333}}{s \cdot \frac{s}{x \cdot -0.3333333333333333}}\right)}\\ \end{array} \]

Alternative 5: 76.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 499999992153374700:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{s + -0.6666666666666666 \cdot \left(s \cdot -3\right)}{s \cdot \left(s \cdot -3\right)}\right)}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 0.20000000298023224)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 499999992153374700.0)
       (/
        1.0
        (+
         1.0
         (+
          1.0
          (*
           x
           (/ (+ s (* -0.6666666666666666 (* s -3.0))) (* s (* s -3.0)))))))
       (if (<= t_0 INFINITY)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/ 1.0 t_0))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.20000000298023224f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 499999992153374700.0f) {
		tmp = 1.0f / (1.0f + (1.0f + (x * ((s + (-0.6666666666666666f * (s * -3.0f))) / (s * (s * -3.0f))))));
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(499999992153374700.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(x * Float32(Float32(s + Float32(Float32(-0.6666666666666666) * Float32(s * Float32(-3.0)))) / Float32(s * Float32(s * Float32(-3.0))))))));
	elseif (t_0 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.20000000298023224))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(499999992153374700.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (x * ((s + (single(-0.6666666666666666) * (s * single(-3.0)))) / (s * (s * single(-3.0)))))));
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 97.2%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s) < 4.99999992e17

   1. Initial program 98.8%

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

      1. distribute-frac-neg 98.8%

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

      2. exp-neg 98.8%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 7.1%

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

      5. sqr-neg 7.1%

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

      6. sqrt-unprod 7.1%

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

      7. add-sqr-sqrt 7.1%

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

      8. add-cube-cbrt 7.1%

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

      9. pow3 7.1%

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

      10. pow-flip 7.1%

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

      11. add-sqr-sqrt 7.1%

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

      12. sqrt-unprod 7.1%

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

      13. sqr-neg 7.1%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 98.9%

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

      16. metadata-eval 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in s around inf 8.4%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 8.4%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 8.4%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]
   9. Step-by-step derivation

      1. +-commutative 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\left(\frac{-0.3333333333333333}{s} + \frac{-0.6666666666666666}{s}\right)}\right)} \]

      2. clear-num 8.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \left(\color{blue}{\frac{1}{\frac{s}{-0.3333333333333333}}} + \frac{-0.6666666666666666}{s}\right)\right)} \]

      3. frac-add 39.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\frac{1 \cdot s + \frac{s}{-0.3333333333333333} \cdot -0.6666666666666666}{\frac{s}{-0.3333333333333333} \cdot s}}\right)} \]

      4. *-un-lft-identity 39.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{\color{blue}{s} + \frac{s}{-0.3333333333333333} \cdot -0.6666666666666666}{\frac{s}{-0.3333333333333333} \cdot s}\right)} \]

      5. div-inv 39.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{s + \color{blue}{\left(s \cdot \frac{1}{-0.3333333333333333}\right)} \cdot -0.6666666666666666}{\frac{s}{-0.3333333333333333} \cdot s}\right)} \]

      6. metadata-eval 39.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{s + \left(s \cdot \color{blue}{-3}\right) \cdot -0.6666666666666666}{\frac{s}{-0.3333333333333333} \cdot s}\right)} \]

      7. div-inv 39.4%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{s + \left(s \cdot -3\right) \cdot -0.6666666666666666}{\color{blue}{\left(s \cdot \frac{1}{-0.3333333333333333}\right)} \cdot s}\right)} \]

      8. metadata-eval 39.4%

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

   10. Applied egg-rr 39.4%

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

   if 4.99999992e17 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 55.1%

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

      1. mul-1-neg 55.1%

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

      2. unsub-neg 55.1%

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

   4. Simplified 55.1%

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

      1. sub-neg 55.1%

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

      2. flip-+ 51.5%

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

      3. metadata-eval 51.5%

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

      4. distribute-neg-frac 51.5%

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

      5. distribute-neg-frac 51.5%

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

      6. distribute-neg-frac 51.5%

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

   6. Applied egg-rr 51.5%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 37.4%

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

      1. mul-1-neg 37.4%

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

      2. unsub-neg 37.4%

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

   4. Simplified 37.4%

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

   5. Taylor expanded in x around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   7. Simplified 19.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-x}{s}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 499999992153374700:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{s + -0.6666666666666666 \cdot \left(s \cdot -3\right)}{s \cdot \left(s \cdot -3\right)}\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 6: 75.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;t_0 \leq 200000000753664:\\ \;\;\;\;\frac{1}{1 + \left(1 - x \cdot \frac{s \cdot -0.3333333333333333 - s \cdot 0.6666666666666666}{s \cdot \left(-s\right)}\right)}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 0.20000000298023224)
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
     (if (<= t_0 200000000753664.0)
       (/
        1.0
        (+
         1.0
         (-
          1.0
          (*
           x
           (/
            (- (* s -0.3333333333333333) (* s 0.6666666666666666))
            (* s (- s)))))))
       (if (<= t_0 INFINITY)
         (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
         (/ 1.0 t_0))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 0.20000000298023224f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else if (t_0 <= 200000000753664.0f) {
		tmp = 1.0f / (1.0f + (1.0f - (x * (((s * -0.3333333333333333f) - (s * 0.6666666666666666f)) / (s * -s)))));
	} else if (t_0 <= ((float) INFINITY)) {
		tmp = 1.0f / ((4.0f - ((x / s) * (x / s))) / ((x / s) + 2.0f));
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.20000000298023224))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	elseif (t_0 <= Float32(200000000753664.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(x * Float32(Float32(Float32(s * Float32(-0.3333333333333333)) - Float32(s * Float32(0.6666666666666666))) / Float32(s * Float32(-s)))))));
	elseif (t_0 <= Float32(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(0.20000000298023224))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	elseif (t_0 <= single(200000000753664.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) - (x * (((s * single(-0.3333333333333333)) - (s * single(0.6666666666666666))) / (s * -s)))));
	elseif (t_0 <= single(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 4 regimes

2. if (/.f32 (neg.f32 x) s) < 0.200000003

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 97.2%

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

   if 0.200000003 < (/.f32 (neg.f32 x) s) < 2.00000001e14

   1. Initial program 98.7%

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

      1. distribute-frac-neg 98.7%

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

      2. exp-neg 98.7%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 7.2%

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

      5. sqr-neg 7.2%

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

      6. sqrt-unprod 7.2%

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

      7. add-sqr-sqrt 7.2%

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

      8. add-cube-cbrt 7.2%

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

      9. pow3 7.2%

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

      10. pow-flip 7.2%

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

      11. add-sqr-sqrt 7.2%

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

      12. sqrt-unprod 7.2%

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

      13. sqr-neg 7.2%

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

      14. sqrt-unprod -0.0%

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

      15. add-sqr-sqrt 98.8%

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

      16. metadata-eval 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in s around inf 8.3%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 8.3%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 8.3%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]
   9. Step-by-step derivation

      1. frac-2neg 8.3%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \left(\color{blue}{\frac{--0.6666666666666666}{-s}} + \frac{-0.3333333333333333}{s}\right)\right)} \]

      2. frac-add 42.2%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\frac{\left(--0.6666666666666666\right) \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}}\right)} \]

      3. metadata-eval 42.2%

         \[\leadsto \frac{1}{1 + \left(1 + x \cdot \frac{\color{blue}{0.6666666666666666} \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}\right)} \]

   10. Applied egg-rr 42.2%

       \[\leadsto \frac{1}{1 + \left(1 + x \cdot \color{blue}{\frac{0.6666666666666666 \cdot s + \left(-s\right) \cdot -0.3333333333333333}{\left(-s\right) \cdot s}}\right)} \]

   if 2.00000001e14 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 52.6%

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

      1. mul-1-neg 52.6%

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

      2. unsub-neg 52.6%

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

   4. Simplified 52.6%

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

      1. sub-neg 52.6%

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

      2. flip-+ 49.3%

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

      3. metadata-eval 49.3%

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

      4. distribute-neg-frac 49.3%

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

      5. distribute-neg-frac 49.3%

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

      6. distribute-neg-frac 49.3%

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

   6. Applied egg-rr 49.3%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 37.4%

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

      1. mul-1-neg 37.4%

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

      2. unsub-neg 37.4%

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

   4. Simplified 37.4%

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

   5. Taylor expanded in x around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   7. Simplified 19.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{-x}{s}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.20000000298023224:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 200000000753664:\\ \;\;\;\;\frac{1}{1 + \left(1 - x \cdot \frac{s \cdot -0.3333333333333333 - s \cdot 0.6666666666666666}{s \cdot \left(-s\right)}\right)}\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 7: 73.1% accurate, 4.0× speedup

Math

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

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 INFINITY)
       (/ 1.0 (/ (- 4.0 (* (/ x s) (/ x s))) (+ (/ x s) 2.0)))
       (/ 1.0 t_0)))))

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

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / 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(Inf))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(Float32(x / s) * Float32(x / s))) / Float32(Float32(x / s) + Float32(2.0))));
	else
		tmp = Float32(Float32(1.0) / t_0);
	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(Inf))
		tmp = single(1.0) / ((single(4.0) - ((x / s) * (x / s))) / ((x / s) + single(2.0)));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.0%

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

   if -0.0199999996 < (/.f32 (neg.f32 x) s) < +inf.0

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 53.9%

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

      1. mul-1-neg 53.9%

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

      2. unsub-neg 53.9%

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

   4. Simplified 53.9%

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

      1. sub-neg 53.9%

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

      2. flip-+ 52.4%

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

      3. metadata-eval 52.4%

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

      4. distribute-neg-frac 52.4%

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

      5. distribute-neg-frac 52.4%

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

      6. distribute-neg-frac 52.4%

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

   6. Applied egg-rr 52.4%

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

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

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 37.4%

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

      1. mul-1-neg 37.4%

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

      2. unsub-neg 37.4%

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

   4. Simplified 37.4%

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

   5. Taylor expanded in x around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. distribute-frac-neg 19.8%

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

   7. Simplified 19.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{elif}\;\frac{-x}{s} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s} \cdot \frac{x}{s}}{\frac{x}{s} + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 8: 75.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -0.019999999552965164)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/
    1.0
    (+
     1.0
     (+ 1.0 (* x (+ (/ -0.6666666666666666 s) (/ -0.3333333333333333 s))))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -0.019999999552965164f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (x * ((-0.6666666666666666f / s) + (-0.3333333333333333f / 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) <= (-0.019999999552965164e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (x * (((-0.6666666666666666e0) / s) + ((-0.3333333333333333e0) / s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-0.019999999552965164))
		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(1.0) + Float32(Float32(1.0) + Float32(x * Float32(Float32(Float32(-0.6666666666666666) / s) + Float32(Float32(-0.3333333333333333) / s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-0.019999999552965164))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (x * ((single(-0.6666666666666666) / s) + (single(-0.3333333333333333) / s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.0%

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

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

   1. Initial program 99.6%

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

      1. distribute-frac-neg 99.6%

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

      2. exp-neg 99.6%

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

      3. add-sqr-sqrt 17.2%

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

      4. sqrt-unprod 31.6%

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

      5. sqr-neg 31.6%

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

      6. sqrt-unprod 15.1%

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

      7. add-sqr-sqrt 31.1%

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

      8. add-cube-cbrt 31.1%

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

      9. pow3 31.1%

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

      10. pow-flip 31.1%

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

      11. add-sqr-sqrt 15.1%

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

      12. sqrt-unprod 31.7%

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

      13. sqr-neg 31.7%

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

      14. sqrt-unprod 17.2%

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

      15. add-sqr-sqrt 99.6%

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

      16. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in s around inf 54.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)\right)}\\ \end{array} \]

Alternative 9: 75.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -200.0)
     (- 1.0 (/ s x))
     (if (<= t_0 1.0) (+ 0.5 (/ (* x 0.25) s)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -200.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f / t_0;
	}
	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 <= (-200.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-200.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-200.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.3%

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

   5. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

   if -200 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

   4. Simplified 98.0%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 36.4%

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

      1. mul-1-neg 36.4%

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

      2. unsub-neg 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in x around inf 36.4%

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

      1. mul-1-neg 36.4%

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

      2. distribute-frac-neg 36.4%

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

   7. Simplified 36.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 10: 75.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -200.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 1.0) (+ 0.5 (/ (* x 0.25) s)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -200.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x * 0.25f) / s);
	} else {
		tmp = 1.0f / t_0;
	}
	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 <= (-200.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x * 0.25e0) / s)
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-200.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x * Float32(0.25)) / s));
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-200.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x * single(0.25)) / s);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.3%

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

   5. Taylor expanded in x around inf 97.3%

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

   if -200 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. associate-*r/ 98.0%

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

   4. Simplified 98.0%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 36.4%

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

      1. mul-1-neg 36.4%

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

      2. unsub-neg 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in x around inf 36.4%

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

      1. mul-1-neg 36.4%

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

      2. distribute-frac-neg 36.4%

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

   7. Simplified 36.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x \cdot 0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 11: 73.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -200.0) (- 1.0 (/ s x)) (if (<= t_0 1.0) 0.5 (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -200.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	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 <= (-200.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-200.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-200.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.3%

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

   5. Taylor expanded in x around inf 97.2%

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

      1. mul-1-neg 97.2%

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

      2. unsub-neg 97.2%

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

   7. Simplified 97.2%

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

   if -200 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 91.0%

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

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

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 36.4%

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

      1. mul-1-neg 36.4%

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

      2. unsub-neg 36.4%

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

   4. Simplified 36.4%

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

   5. Taylor expanded in x around inf 36.4%

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

      1. mul-1-neg 36.4%

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

      2. distribute-frac-neg 36.4%

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

   7. Simplified 36.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 12: 74.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{-1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -200.0)
   (/ 1.0 (+ 1.0 (/ s x)))
   (/ 1.0 (+ 1.0 (+ 1.0 (* x (/ -1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -200.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (x * (-1.0f / 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) <= (-200.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (x * ((-1.0e0) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-200.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(x * Float32(Float32(-1.0) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-200.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (x * (single(-1.0) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.3%

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

   5. Taylor expanded in x around inf 97.3%

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

   if -200 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

      1. distribute-frac-neg 99.6%

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

      2. exp-neg 99.6%

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

      3. add-sqr-sqrt 17.7%

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

      4. sqrt-unprod 31.7%

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

      5. sqr-neg 31.7%

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

      6. sqrt-unprod 15.0%

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

      7. add-sqr-sqrt 31.2%

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

      8. add-cube-cbrt 31.2%

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

      9. pow3 31.2%

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

      10. pow-flip 31.2%

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

      11. add-sqr-sqrt 15.0%

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

      12. sqrt-unprod 31.8%

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

      13. sqr-neg 31.8%

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

      14. sqrt-unprod 17.7%

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

      15. add-sqr-sqrt 99.6%

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

      16. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in s around inf 54.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]

   9. Taylor expanded in s around 0 54.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{-1}{s}\right)}\\ \end{array} \]

Alternative 13: 75.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{-1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -0.019999999552965164)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 1.0 (+ 1.0 (* x (/ -1.0 s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -0.019999999552965164f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (x * (-1.0f / 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) <= (-0.019999999552965164e0)) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (x * ((-1.0e0) / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-0.019999999552965164))
		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(1.0) + Float32(Float32(1.0) + Float32(x * Float32(Float32(-1.0) / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-0.019999999552965164))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (x * (single(-1.0) / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.0%

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

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

   1. Initial program 99.6%

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

      1. distribute-frac-neg 99.6%

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

      2. exp-neg 99.6%

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

      3. add-sqr-sqrt 17.2%

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

      4. sqrt-unprod 31.6%

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

      5. sqr-neg 31.6%

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

      6. sqrt-unprod 15.1%

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

      7. add-sqr-sqrt 31.1%

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

      8. add-cube-cbrt 31.1%

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

      9. pow3 31.1%

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

      10. pow-flip 31.1%

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

      11. add-sqr-sqrt 15.1%

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

      12. sqrt-unprod 31.7%

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

      13. sqr-neg 31.7%

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

      14. sqrt-unprod 17.2%

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

      15. add-sqr-sqrt 99.6%

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

      16. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

   4. Taylor expanded in s around inf 54.0%

      \[\leadsto \frac{1}{1 + \color{blue}{\left(1 + \left(-0.6666666666666666 \cdot \frac{x}{s} + -0.3333333333333333 \cdot \frac{x}{s}\right)\right)}} \]
   5. Step-by-step derivation

      1. +-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{x}{s} + -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      2. *-commutative 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \left(\color{blue}{\frac{x}{s} \cdot -0.3333333333333333} + -0.6666666666666666 \cdot \frac{x}{s}\right)\right)} \]

      3. fma-def 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, -0.6666666666666666 \cdot \frac{x}{s}\right)}\right)} \]

      4. associate-*r/ 54.0%

         \[\leadsto \frac{1}{1 + \left(1 + \mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \color{blue}{\frac{-0.6666666666666666 \cdot x}{s}}\right)\right)} \]

   6. Applied egg-rr 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{\mathsf{fma}\left(\frac{x}{s}, -0.3333333333333333, \frac{-0.6666666666666666 \cdot x}{s}\right)}\right)} \]

   8. Simplified 54.0%

      \[\leadsto \frac{1}{1 + \left(1 + \color{blue}{x \cdot \left(\frac{-0.6666666666666666}{s} + \frac{-0.3333333333333333}{s}\right)}\right)} \]

   9. Taylor expanded in s around 0 54.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -0.019999999552965164:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + x \cdot \frac{-1}{s}\right)}\\ \end{array} \]

Alternative 14: 74.4% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -200.0) (/ 1.0 (+ 1.0 (/ s x))) (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -200.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-200.0e0)) then
        tmp = 1.0e0 / (1.0e0 + (s / x))
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-200.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-200.0))
		tmp = single(1.0) / (single(1.0) + (s / x));
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < -200

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 97.3%

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

   5. Taylor expanded in x around inf 97.3%

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

   if -200 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 54.0%

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

      1. mul-1-neg 54.0%

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

      2. unsub-neg 54.0%

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

   4. Simplified 54.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -200:\\ \;\;\;\;\frac{1}{1 + \frac{s}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]

Alternative 15: 68.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0015999999595806003:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0015999999595806003)
   (/ (- s) x)
   (if (<= x 2.00000009162741e-18) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0015999999595806003f) {
		tmp = -s / x;
	} else if (x <= 2.00000009162741e-18f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0015999999595806003e0)) then
        tmp = -s / x
    else if (x <= 2.00000009162741e-18) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0015999999595806003))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(2.00000009162741e-18))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0015999999595806003))
		tmp = -s / x;
	elseif (x <= single(2.00000009162741e-18))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00159999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

      1. mul-1-neg 51.1%

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

      2. unsub-neg 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. neg-mul-1 45.8%

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

   7. Simplified 45.8%

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

   if -0.00159999996 < x < 2.00000009e-18

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 49.7%

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

   if 2.00000009e-18 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 98.1%

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

   5. Taylor expanded in x around inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. unsub-neg 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0015999999595806003:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 2.00000009162741 \cdot 10^{-18}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]

Alternative 16: 47.0% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0015999999595806003:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0015999999595806003) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0015999999595806003f) {
		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 <= (-0.0015999999595806003e0)) 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(-0.0015999999595806003))
		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(-0.0015999999595806003))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -0.00159999996

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

      1. mul-1-neg 51.1%

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

      2. unsub-neg 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in x around inf 45.8%

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

      1. associate-*r/ 45.8%

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

      2. neg-mul-1 45.8%

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

   7. Simplified 45.8%

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

   if -0.00159999996 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 40.8%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0015999999595806003:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 17: 35.6% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 30.4%

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

3. Final simplification 30.4%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))