Logistic function

Percentage Accurate: 99.8% → 99.7%

Time: 9.9s

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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-2} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (hypot 1.0 (pow (exp 3.0) (* (/ x s) -0.16666666666666666))) -2.0))

C

float code(float x, float s) {
	return powf(hypotf(1.0f, powf(expf(3.0f), ((x / s) * -0.16666666666666666f))), -2.0f);
}

Julia

function code(x, s)
	return hypot(Float32(1.0), (exp(Float32(3.0)) ^ Float32(Float32(x / s) * Float32(-0.16666666666666666)))) ^ Float32(-2.0)
end

MATLAB

function tmp = code(x, s)
	tmp = hypot(single(1.0), (exp(single(3.0)) ^ ((x / s) * single(-0.16666666666666666)))) ^ single(-2.0);
end

Derivation

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 99.5%

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

   3. div-inv 99.5%

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

   4. add-sqr-sqrt 54.2%

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

   5. sqrt-unprod 63.3%

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

   6. sqr-neg 63.3%

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

   7. sqrt-unprod 11.1%

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

   8. add-sqr-sqrt 26.3%

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

   9. div-inv 26.3%

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

   10. pow1 26.3%

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

   11. pow1 26.3%

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

   12. add-cbrt-cube 26.3%

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

   13. pow1/3 26.3%

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

   14. pow-flip 26.3%

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

3. Applied egg-rr 99.1%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-div 98.5%

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

   3. metadata-eval 98.5%

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

   4. sqr-pow 98.5%

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

   5. hypot-1-def 98.5%

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

   6. exp-prod 98.5%

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

   7. pow-pow 98.5%

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

   8. metadata-eval 98.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot \color{blue}{-0.16666666666666666}\right)}\right)} \cdot \sqrt{\frac{1}{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]

   9. sqrt-div 98.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}}} \]

   10. metadata-eval 98.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]

   11. sqr-pow 98.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{1}{\sqrt{1 + \color{blue}{{\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)} \cdot {\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)}} \]
6. Step-by-step derivation

   1. unpow-1 99.2%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1}} \cdot \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \]

   2. unpow-1 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1} \cdot \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1}} \]

   3. pow-sqr 99.7%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{\left(2 \cdot -1\right)}} \]

   4. metadata-eval 99.7%

      \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{\color{blue}{-2}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-2}} \]

8. Final simplification 99.7%

   \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-2} \]

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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 99.5%

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

   3. div-inv 99.5%

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

   4. add-sqr-sqrt 54.2%

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

   5. sqrt-unprod 63.3%

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

   6. sqr-neg 63.3%

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

   7. sqrt-unprod 11.1%

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

   8. add-sqr-sqrt 26.3%

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

   9. div-inv 26.3%

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

   10. pow1 26.3%

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

   11. pow1 26.3%

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

   12. add-cbrt-cube 26.3%

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

   13. pow1/3 26.3%

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

   14. pow-flip 26.3%

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

3. Applied egg-rr 99.1%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.5%

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

   2. sqrt-div 98.5%

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

   3. metadata-eval 98.5%

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

   4. sqr-pow 98.5%

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

   5. hypot-1-def 98.5%

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

   6. exp-prod 98.5%

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

   7. pow-pow 98.5%

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

   8. metadata-eval 98.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot \color{blue}{-0.16666666666666666}\right)}\right)} \cdot \sqrt{\frac{1}{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]

   9. sqrt-div 98.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}}} \]

   10. metadata-eval 98.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{\color{blue}{1}}{\sqrt{1 + {\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]

   11. sqr-pow 98.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{1}{\sqrt{1 + \color{blue}{{\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)} \cdot {\left(e^{3 \cdot \frac{x}{s}}\right)}^{\left(\frac{-0.3333333333333333}{2}\right)}}}} \]

5. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \cdot \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)}} \]
6. Step-by-step derivation

   1. unpow-1 99.2%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1}} \cdot \frac{1}{\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)} \]

   2. unpow-1 99.2%

      \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1} \cdot \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-1}} \]

   3. pow-sqr 99.7%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{\left(2 \cdot -1\right)}} \]

   4. metadata-eval 99.7%

      \[\leadsto {\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{\color{blue}{-2}} \]

7. Simplified 99.7%

   \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.16666666666666666\right)}\right)\right)}^{-2}} \]

8. Taylor expanded in x around inf 99.7%

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

9. Final simplification 99.7%

   \[\leadsto {\left(\mathsf{hypot}\left(1, e^{\frac{x}{s} \cdot -0.5}\right)\right)}^{-2} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.3333333333333333\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp 3.0) (* (/ x s) -0.3333333333333333)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(3.0f), ((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(3.0e0) ** ((x / s) * (-0.3333333333333333e0))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(3.0)) ^ Float32(Float32(x / s) * Float32(-0.3333333333333333)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(3.0)) ^ ((x / s) * single(-0.3333333333333333))));
end

Derivation

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 99.5%

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

   3. div-inv 99.5%

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

   4. add-sqr-sqrt 54.2%

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

   5. sqrt-unprod 63.3%

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

   6. sqr-neg 63.3%

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

   7. sqrt-unprod 11.1%

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

   8. add-sqr-sqrt 26.3%

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

   9. div-inv 26.3%

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

   10. pow1 26.3%

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

   11. pow1 26.3%

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

   12. add-cbrt-cube 26.3%

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

   13. pow1/3 26.3%

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

   14. pow-flip 26.3%

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

3. Applied egg-rr 99.1%

   \[\leadsto \frac{1}{1 + \color{blue}{{\left(e^{3 \cdot \frac{x}{s}}\right)}^{-0.3333333333333333}}} \]
4. Step-by-step derivation

   1. *-un-lft-identity 99.1%

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

   2. exp-prod 99.1%

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

   3. pow-pow 99.6%

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

5. Applied egg-rr 99.6%

   \[\leadsto \frac{1}{1 + \color{blue}{1 \cdot {\left(e^{3}\right)}^{\left(\frac{x}{s} \cdot -0.3333333333333333\right)}}} \]
6. Step-by-step derivation

   1. *-lft-identity 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. div-inv 99.6%

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

   2. exp-prod 80.7%

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

   3. neg-mul-1 80.7%

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

   4. exp-prod 80.7%

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

   5. pow-pow 99.6%

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

   6. div-inv 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 6: 62.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot \left(-s\right)}\right) - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 1.9999999996399175e-23)
   0.5
   (/ 1.0 (+ 2.0 (- (* 0.5 (* (* x x) (/ -1.0 (* s (- s))))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 1.9999999996399175e-23f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * ((x * x) * (-1.0f / (s * -s)))) - (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 <= 1.9999999996399175e-23) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * ((x * x) * ((-1.0e0) / (s * -s)))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(1.9999999996399175e-23))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(Float32(x * x) * Float32(Float32(-1.0) / Float32(s * Float32(-s))))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(1.9999999996399175e-23))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * ((x * x) * (single(-1.0) / (s * -s)))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 2e-23

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 48.3%

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

   if 2e-23 < (neg.f32 x)

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 81.6%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. unpow2 81.6%

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

      5. unpow2 81.6%

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

      6. times-frac 71.5%

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

   4. Simplified 71.5%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right) - \frac{x}{s}\right)}} \]
   5. Step-by-step derivation

      1. frac-times 81.6%

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

   6. Applied egg-rr 81.6%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\frac{x \cdot x}{s \cdot s}} - \frac{x}{s}\right)} \]
   7. Step-by-step derivation

      1. frac-2neg 81.6%

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

      2. div-inv 85.2%

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

   8. Applied egg-rr 85.2%

      \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\left(\left(-x \cdot x\right) \cdot \frac{1}{-s \cdot s}\right)} - \frac{x}{s}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-neg-in 85.2%

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

      2. distribute-rgt-neg-in 85.2%

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

   10. Simplified 85.2%

       \[\leadsto \frac{1}{2 + \left(0.5 \cdot \color{blue}{\left(\left(x \cdot \left(-x\right)\right) \cdot \frac{1}{s \cdot \left(-s\right)}\right)} - \frac{x}{s}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 1.9999999996399175 \cdot 10^{-23}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{s \cdot \left(-s\right)}\right) - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 7: 62.1% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * (((s * s) / x) * (1.0f / 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 / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (((s * s) / x) * (1.0e0 / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(Float32(s * s) / x) * Float32(Float32(1.0) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (((s * s) / x) * (single(1.0) / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 76.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. unpow2 76.8%

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

      5. unpow2 76.8%

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

      6. times-frac 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. unpow2 75.9%

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

      2. unpow2 75.9%

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

   7. Simplified 75.9%

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

      1. associate-/r* 80.3%

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

      2. div-inv 80.3%

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

   9. Applied egg-rr 80.3%

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

4. Final simplification 61.6%

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

Alternative 8: 58.0% accurate, 7.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * (s * ((s / x) / 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 / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (s * ((s / x) / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(Float32(s / x) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (s * ((s / x) / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 76.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. unpow2 76.8%

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

      5. unpow2 76.8%

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

      6. times-frac 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. unpow2 75.9%

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

      2. unpow2 75.9%

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

   7. Simplified 75.9%

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

   8. Taylor expanded in s around 0 75.9%

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

      1. unpow2 75.9%

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

      2. associate-*r/ 64.7%

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

      3. unpow2 64.7%

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

      4. associate-/r* 64.9%

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

   10. Simplified 64.9%

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

4. Final simplification 56.1%

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

Alternative 9: 58.0% accurate, 7.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * ((s / x) * (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 / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s / x) * (s / x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * Float32(s / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s / x) * (s / x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 76.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. unpow2 76.8%

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

      5. unpow2 76.8%

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

      6. times-frac 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. unpow2 75.9%

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

      2. unpow2 75.9%

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

      3. times-frac 64.9%

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

   7. Simplified 64.9%

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

4. Final simplification 56.1%

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

Alternative 10: 58.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-(x / s) <= 1.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * (s / (x * (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * (s / (x * (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 76.8%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. unpow2 76.8%

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

      5. unpow2 76.8%

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

      6. times-frac 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in x around inf 75.9%

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

      1. unpow2 75.9%

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

      2. unpow2 75.9%

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

   7. Simplified 75.9%

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

      1. associate-/l* 65.0%

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

      2. div-inv 65.0%

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

   9. Applied egg-rr 65.0%

      \[\leadsto 2 \cdot \color{blue}{\left(s \cdot \frac{1}{\frac{x \cdot x}{s}}\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ 65.0%

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

       2. *-rgt-identity 65.0%

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

       3. *-lft-identity 65.0%

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

       4. times-frac 65.2%

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

       5. /-rgt-identity 65.2%

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

   11. Simplified 65.2%

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

4. Final simplification 56.1%

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

Alternative 11: 60.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 100000000.0) 0.5 (* 2.0 (/ (* s s) (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 100000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 2.0f * ((s * s) / (x * 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 / s) <= 100000000.0e0) then
        tmp = 0.5e0
    else
        tmp = 2.0e0 * ((s * s) / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(100000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(s * s) / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(100000000.0))
		tmp = single(0.5);
	else
		tmp = single(2.0) * ((s * s) / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1e8

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 47.5%

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

   if 1e8 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 90.1%

      \[\leadsto \frac{1}{\color{blue}{2 + \left(-1 \cdot \frac{x}{s} + 0.5 \cdot \frac{{x}^{2}}{{s}^{2}}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 90.1%

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

      2. mul-1-neg 90.1%

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

      3. unsub-neg 90.1%

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

      4. unpow2 90.1%

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

      5. unpow2 90.1%

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

      6. times-frac 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in x around inf 89.2%

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

      1. unpow2 89.2%

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

      2. unpow2 89.2%

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

   7. Simplified 89.2%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \end{array} \]

Alternative 12: 49.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(-1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.2%

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

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

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0 60.6%

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

      1. mul-1-neg 60.6%

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

      2. unsub-neg 60.6%

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

   4. Simplified 60.6%

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

4. Final simplification 47.6%

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

Alternative 13: 48.1% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (- (/ x s)))) (if (<= t_0 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 <= 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 <= 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(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(1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.1%

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

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

   1. Initial program 99.0%

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

   2. Taylor expanded in x around 0 37.1%

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

      1. mul-1-neg 37.1%

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

      2. unsub-neg 37.1%

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

   4. Simplified 37.1%

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

   5. Taylor expanded in x around inf 37.0%

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

      1. neg-mul-1 37.0%

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

      2. distribute-neg-frac 37.0%

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

   7. Simplified 37.0%

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

4. Final simplification 46.0%

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

Alternative 14: 47.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00039999998989515007) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00039999998989515007f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.00039999998989515007e0)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.00039999998989515007))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.00039999998989515007))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999e-4

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in x around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. neg-mul-1 44.7%

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

   7. Simplified 44.7%

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

      1. div-inv 44.7%

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

   9. Applied egg-rr 44.7%

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

       1. add-sqr-sqrt 44.7%

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

       2. sqrt-unprod 82.3%

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

       3. un-div-inv 82.3%

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

       4. un-div-inv 82.3%

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

       5. frac-times 82.3%

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

       6. sqr-neg 82.3%

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

       7. clear-num 86.6%

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

       8. sqrt-div 86.6%

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

       9. metadata-eval 86.6%

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

       10. times-frac 86.6%

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

       11. sqrt-prod -0.0%

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

       12. add-sqr-sqrt 50.4%

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

   11. Applied egg-rr 50.4%

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

   if -3.9999999e-4 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 44.4%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 15: 46.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00039999998989515007) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00039999998989515007f) {
		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 <= (-0.00039999998989515007e0)) 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(-0.00039999998989515007))
		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(-0.00039999998989515007))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999e-4

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in x around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. neg-mul-1 44.7%

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

   7. Simplified 44.7%

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

   if -3.9999999e-4 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 44.4%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 16: 46.6% accurate, 21.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00039999998989515007) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00039999998989515007f) {
		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 <= (-0.00039999998989515007e0)) 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(-0.00039999998989515007))
		tmp = Float32(s / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.00039999998989515007))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999e-4

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in x around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. neg-mul-1 44.7%

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

   7. Simplified 44.7%

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

      1. div-inv 44.7%

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

   9. Applied egg-rr 44.7%

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

       1. expm1-log1p-u 44.7%

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

       2. expm1-udef 94.0%

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

       3. add-sqr-sqrt 94.0%

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

       4. sqrt-unprod 94.0%

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

       5. un-div-inv 94.0%

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

       6. un-div-inv 94.0%

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

       7. frac-times 94.0%

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

       8. sqr-neg 94.0%

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

       9. times-frac 94.0%

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

       10. sqrt-prod 27.4%

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

       11. add-sqr-sqrt 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. expm1-def 44.7%

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

       2. expm1-log1p 44.7%

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

   13. Simplified 44.7%

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

   if -3.9999999e-4 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 44.4%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00039999998989515007:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 17: 35.3% 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 99.6%

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

2. Taylor expanded in x around 0 35.1%

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

3. Final simplification 35.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))