Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 12.9s

Alternatives: 20

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{-\frac{x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (- (/ x s)))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf(-(x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(-Float32(x / s))))))
end

Derivation

1. Initial program 99.8%

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

   1. inv-pow N/A

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

   2. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. log-pow N/A

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

   5. inv-pow N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. log-rec N/A

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

   8. neg-lowering-neg.f32 N/A

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

   9. accelerator-lowering-log1p.f32 N/A

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

   10. exp-lowering-exp.f32 N/A

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

   11. distribute-frac-neg N/A

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

   12. neg-lowering-neg.f32 N/A

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

   13. /-lowering-/.f32 99.9

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

4. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left({\left(e \cdot e\right)}^{\left(-2 \cdot \left(\frac{x}{s} \cdot 0.16666666666666666\right)\right)}, e^{\frac{x}{s} \cdot -0.3333333333333333}, 1\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/
  1.0
  (fma
   (pow (* E E) (- (* 2.0 (* (/ x s) 0.16666666666666666))))
   (exp (* (/ x s) -0.3333333333333333))
   1.0)))

C

float code(float x, float s) {
	return 1.0f / fmaf(powf((((float) M_E) * ((float) M_E)), -(2.0f * ((x / s) * 0.16666666666666666f))), expf(((x / s) * -0.3333333333333333f)), 1.0f);
}

Julia

function code(x, s)
	return Float32(Float32(1.0) / fma((Float32(Float32(exp(1)) * Float32(exp(1))) ^ Float32(-Float32(Float32(2.0) * Float32(Float32(x / s) * Float32(0.16666666666666666))))), exp(Float32(Float32(x / s) * Float32(-0.3333333333333333))), Float32(1.0)))
end

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f32 N/A

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

   4. exp-1-e N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   8. /-lowering-/.f32 99.7

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

4. Applied egg-rr 99.7%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\frac{1}{3}}\right)}}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\left(\frac{1}{3} \cdot 1\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   4. log-E N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   5. log-pow N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\log \left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   6. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   7. pow-pow N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   8. pow-lowering-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   11. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\color{blue}{\mathsf{E}\left(\right)} \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \color{blue}{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   13. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\color{blue}{\mathsf{E}\left(\right)} \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   14. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \color{blue}{\mathsf{E}\left(\right)}\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   16. pow1/3 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \color{blue}{\left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   17. log-pow N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\left(\frac{1}{3} \cdot \log \mathsf{E}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   18. log-E N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\left(\frac{1}{3} \cdot \color{blue}{1}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\frac{1}{3}} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   20. neg-lowering-neg.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}\right)}} \]

   21. /-lowering-/.f32 99.8

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

6. Applied egg-rr 99.8%

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

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)} + 1}} \]

   2. sqr-pow N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}} + 1} \]

   3. pow-sqr N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}} + 1} \]

   4. associate-*r* N/A

      \[\leadsto \frac{1}{{\color{blue}{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} + 1} \]

   5. unpow-prod-down N/A

      \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot {\mathsf{E}\left(\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}} + 1} \]

   6. pow-pow N/A

      \[\leadsto \frac{1}{{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot \color{blue}{{\left({\mathsf{E}\left(\right)}^{2}\right)}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}} + 1} \]

   7. pow2 N/A

      \[\leadsto \frac{1}{{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} + 1} \]

   8. unpow-prod-down N/A

      \[\leadsto \frac{1}{{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot \color{blue}{\left({\mathsf{E}\left(\right)}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot {\mathsf{E}\left(\right)}^{\left(\frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}\right)} + 1} \]

   9. sqr-pow N/A

      \[\leadsto \frac{1}{{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)} \cdot \color{blue}{{\mathsf{E}\left(\right)}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} + 1} \]

   10. accelerator-lowering-fma.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(2 \cdot \frac{\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}{2}\right)}, {\mathsf{E}\left(\right)}^{\left(\frac{1}{3} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}, 1\right)}} \]

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{1}{\mathsf{fma}\left({\left(e \cdot e\right)}^{\left(-2 \cdot \left(\frac{x}{s} \cdot 0.16666666666666666\right)\right)}, e^{\frac{x}{s} \cdot -0.3333333333333333}, 1\right)} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(Float32(exp(1)) * Float32(Float32(exp(1)) * Float32(exp(1)))) ^ Float32(Float32(x / s) * Float32(-Float32(0.3333333333333333))))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((single(2.71828182845904523536) * (single(2.71828182845904523536) * single(2.71828182845904523536))) ^ ((x / s) * -single(0.3333333333333333))));
end

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f32 N/A

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

   4. exp-1-e N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   8. /-lowering-/.f32 99.7

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

4. Applied egg-rr 99.7%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\frac{1}{3}}\right)}}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\left(\frac{1}{3} \cdot 1\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   4. log-E N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   5. log-pow N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\log \left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   6. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   7. pow-pow N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   8. pow-lowering-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   11. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\color{blue}{\mathsf{E}\left(\right)} \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \color{blue}{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   13. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\color{blue}{\mathsf{E}\left(\right)} \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   14. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \color{blue}{\mathsf{E}\left(\right)}\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   16. pow1/3 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \color{blue}{\left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   17. log-pow N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\left(\frac{1}{3} \cdot \log \mathsf{E}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   18. log-E N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\left(\frac{1}{3} \cdot \color{blue}{1}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\frac{1}{3}} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   20. neg-lowering-neg.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}\right)}} \]

   21. /-lowering-/.f32 99.8

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(Float32(exp(1)) * Float32(Float32(exp(1)) * Float32(exp(1)))) ^ Float32(Float32(x * Float32(-0.3333333333333333)) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + ((single(2.71828182845904523536) * (single(2.71828182845904523536) * single(2.71828182845904523536))) ^ ((x * single(-0.3333333333333333)) / s)));
end

Derivation

1. Initial program 99.8%

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

   1. *-lft-identity N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f32 N/A

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

   4. exp-1-e N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   5. E-lowering-E.f32 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}} \]

   6. distribute-frac-neg N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   7. neg-lowering-neg.f32 N/A

      \[\leadsto \frac{1}{1 + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}}} \]

   8. /-lowering-/.f32 99.7

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

4. Applied egg-rr 99.7%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\frac{1}{3}}\right)}}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\left(\frac{1}{3} \cdot 1\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   4. log-E N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\log \mathsf{E}\left(\right)}\right)}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   5. log-pow N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\color{blue}{\log \left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   6. pow1/3 N/A

      \[\leadsto \frac{1}{1 + {\left({\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\log \color{blue}{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}}\right)}^{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}} \]

   7. pow-pow N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   8. pow-lowering-pow.f32 N/A

      \[\leadsto \frac{1}{1 + \color{blue}{{\left(\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right) \cdot \mathsf{E}\left(\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\color{blue}{\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   11. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\color{blue}{\mathsf{E}\left(\right)} \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   12. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \color{blue}{\left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)}\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   13. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\color{blue}{\mathsf{E}\left(\right)} \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   14. E-lowering-E.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \color{blue}{\mathsf{E}\left(\right)}\right)\right)}^{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   15. *-lowering-*.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}}} \]

   16. pow1/3 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\log \color{blue}{\left({\mathsf{E}\left(\right)}^{\frac{1}{3}}\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   17. log-pow N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\left(\frac{1}{3} \cdot \log \mathsf{E}\left(\right)\right)} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   18. log-E N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\left(\frac{1}{3} \cdot \color{blue}{1}\right) \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   19. metadata-eval N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\color{blue}{\frac{1}{3}} \cdot \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)}} \]

   20. neg-lowering-neg.f32 N/A

       \[\leadsto \frac{1}{1 + {\left(\mathsf{E}\left(\right) \cdot \left(\mathsf{E}\left(\right) \cdot \mathsf{E}\left(\right)\right)\right)}^{\left(\frac{1}{3} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{x}{s}\right)\right)}\right)}} \]

   21. /-lowering-/.f32 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-lowering-*.f32 99.8

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 6: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -4.0)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ (/ x s) s) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -4.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf(((x / s) / s), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(Float32(x / s) / s), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   5. Simplified 86.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{s}}{s}}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), 2\right)} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{s}}{s}}, \mathsf{fma}\left(\frac{-1}{6}, \frac{x}{s}, \frac{1}{2}\right), \frac{-1}{s}\right), 2\right)} \]

      3. /-lowering-/.f32 92.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\color{blue}{\frac{x}{s}}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)} \]

   7. Applied egg-rr 92.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{\frac{x}{s}}{s}}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s}, \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -4.0)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ x s) (/ (fma (/ x s) -0.16666666666666666 0.5) s) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -4.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / s), (fmaf((x / s), -0.16666666666666666f, 0.5f) / s), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / s), Float32(fma(Float32(x / s), Float32(-0.16666666666666666), Float32(0.5)) / s), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   5. Simplified 86.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{x \cdot \left(\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}\right)}{s \cdot s}} + \frac{-1}{s}, 2\right)} \]

      2. times-frac N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\frac{x}{s} \cdot \frac{\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}}{s}} + \frac{-1}{s}, 2\right)} \]

      3. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}}{s}, \frac{-1}{s}\right)}, 2\right)} \]

      4. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}}{s}, \frac{-1}{s}\right), 2\right)} \]

      5. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \color{blue}{\frac{\frac{-1}{6} \cdot \frac{x}{s} + \frac{1}{2}}{s}}, \frac{-1}{s}\right), 2\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{\frac{x}{s} \cdot \frac{-1}{6}} + \frac{1}{2}}{s}, \frac{-1}{s}\right), 2\right)} \]

      7. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{-1}{6}, \frac{1}{2}\right)}}{s}, \frac{-1}{s}\right), 2\right)} \]

      8. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\mathsf{fma}\left(\color{blue}{\frac{x}{s}}, \frac{-1}{6}, \frac{1}{2}\right)}{s}, \frac{-1}{s}\right), 2\right)} \]

      9. /-lowering-/.f32 92.7

         \[\leadsto \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s}, \color{blue}{\frac{-1}{s}}\right), 2\right)} \]

   7. Applied egg-rr 92.7%

      \[\leadsto \frac{1}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(\frac{x}{s}, \frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s}, \frac{-1}{s}\right)}, 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s}, \frac{\mathsf{fma}\left(\frac{x}{s}, -0.16666666666666666, 0.5\right)}{s}, \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 20.0)
   0.5
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.0%

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

   if 20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right) + 2}} \]

      2. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}, 2\right)}} \]

   5. Simplified 92.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -4.0)
   0.5
   (/ 1.0 (fma x (/ (fma (/ x s) 0.5 -1.0) s) 2.0))))

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(-4.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, Float32(fma(Float32(x / s), Float32(0.5), Float32(-1.0)) / s), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 81.7%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(0.5, \frac{x}{s}, -1\right), 2\right)}} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f32 N/A

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

      4. /-lowering-/.f32 N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f32 N/A

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

      7. /-lowering-/.f32 86.9

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

   7. Applied egg-rr 86.9%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\frac{x}{s}, 0.5, -1\right)}{s}, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 20.0) 0.5 (/ 1.0 (* 0.5 (* x (/ x (* s s)))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 20.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (0.5f * (x * (x / (s * 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) <= 20.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (0.5e0 * (x * (x / (s * s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(20.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(0.5) * Float32(x * Float32(x / Float32(s * s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(20.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(0.5) * (x * (x / (s * s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.0%

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

   if 20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 81.9%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in s around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot s\right)}}{{s}^{3}}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}}{{s}^{3}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{3} \cdot \frac{-1}{6}} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}}{{s}^{3}}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      17. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      19. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      20. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      21. *-lowering-*.f32 84.4

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   8. Simplified 84.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]

   9. Taylor expanded in x around 0

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

       1. associate-*r/ N/A

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

       2. *-rgt-identity N/A

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

       3. metadata-eval N/A

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

       4. log-E N/A

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

       5. associate-*r* N/A

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

       6. associate-*r/ N/A

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

       7. *-lowering-*.f32 N/A

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

       8. log-E N/A

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

       9. metadata-eval N/A

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

       10. *-rgt-identity N/A

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

       11. unpow2 N/A

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

       12. associate-/l* N/A

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

       13. *-lowering-*.f32 N/A

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

       14. /-lowering-/.f32 N/A

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

       15. unpow2 N/A

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

       16. *-lowering-*.f32 85.7

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

   11. Simplified 85.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 20:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 10000.0) 0.5 (/ (* (* s (* s s)) -6.0) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 10000.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * (s * s)) * -6.0f) / (x * (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) <= 10000.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * (s * s)) * (-6.0e0)) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * Float32(s * s)) * Float32(-6.0)) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(10000.0))
		tmp = single(0.5);
	else
		tmp = ((s * (s * s)) * single(-6.0)) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1e4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.2%

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

   if 1e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 86.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      2. /-lowering-/.f32 N/A

         \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{{s}^{3} \cdot -6}}{{x}^{3}} \]

      5. cube-mult N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot \left(s \cdot s\right)\right)} \cdot -6}{{x}^{3}} \]

      6. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{{s}^{2}}\right) \cdot -6}{{x}^{3}} \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot {s}^{2}\right)} \cdot -6}{{x}^{3}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{\left(s \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot -6}{{x}^{3}} \]

      10. cube-mult N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{{x}^{2}}} \]

      12. *-lowering-*.f32 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{\color{blue}{x \cdot {x}^{2}}} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

      14. *-lowering-*.f32 88.5

          \[\leadsto \frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   8. Simplified 88.5%

      \[\leadsto \color{blue}{\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot \left(s \cdot s\right)\right) \cdot -6}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 10000.0) 0.5 (/ (* s (* (* s s) -6.0)) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 10000.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * ((s * s) * -6.0f)) / (x * (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) <= 10000.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * ((s * s) * (-6.0e0))) / (x * (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * Float32(Float32(s * s) * Float32(-6.0))) / Float32(x * Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(10000.0))
		tmp = single(0.5);
	else
		tmp = (s * ((s * s) * single(-6.0))) / (x * (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1e4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.2%

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

   if 1e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 86.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in s around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot s\right)}}{{s}^{3}}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}}{{s}^{3}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{3} \cdot \frac{-1}{6}} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}}{{s}^{3}}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      17. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      19. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      20. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      21. *-lowering-*.f32 89.4

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   8. Simplified 89.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

       2. /-lowering-/.f32 N/A

          \[\leadsto \color{blue}{\frac{-6 \cdot {s}^{3}}{{x}^{3}}} \]

       3. unpow3 N/A

          \[\leadsto \frac{-6 \cdot \color{blue}{\left(\left(s \cdot s\right) \cdot s\right)}}{{x}^{3}} \]

       4. unpow2 N/A

          \[\leadsto \frac{-6 \cdot \left(\color{blue}{{s}^{2}} \cdot s\right)}{{x}^{3}} \]

       5. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(-6 \cdot {s}^{2}\right) \cdot s}}{{x}^{3}} \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \frac{\color{blue}{\left(-6 \cdot {s}^{2}\right) \cdot s}}{{x}^{3}} \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \frac{\color{blue}{\left(-6 \cdot {s}^{2}\right)} \cdot s}{{x}^{3}} \]

       8. unpow2 N/A

          \[\leadsto \frac{\left(-6 \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot s}{{x}^{3}} \]

       9. *-lowering-*.f32 N/A

          \[\leadsto \frac{\left(-6 \cdot \color{blue}{\left(s \cdot s\right)}\right) \cdot s}{{x}^{3}} \]

       10. cube-mult N/A

           \[\leadsto \frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       11. unpow2 N/A

           \[\leadsto \frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{x \cdot \color{blue}{{x}^{2}}} \]

       12. *-lowering-*.f32 N/A

           \[\leadsto \frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{\color{blue}{x \cdot {x}^{2}}} \]

       13. unpow2 N/A

           \[\leadsto \frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

       14. *-lowering-*.f32 88.5

           \[\leadsto \frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{x \cdot \color{blue}{\left(x \cdot x\right)}} \]

   11. Simplified 88.5%

       \[\leadsto \color{blue}{\frac{\left(-6 \cdot \left(s \cdot s\right)\right) \cdot s}{x \cdot \left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{s}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 10000.0)
   0.5
   (* (* s s) (/ s (* -0.16666666666666666 (* x (* x x)))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 10000.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * s) * (s / (-0.16666666666666666f * (x * (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) <= 10000.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * s) * (s / ((-0.16666666666666666e0) * (x * (x * x))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * s) * Float32(s / Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(10000.0))
		tmp = single(0.5);
	else
		tmp = (s * s) * (s / (single(-0.16666666666666666) * (x * (x * x))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 1e4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.2%

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

   if 1e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 86.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in s around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot s\right)}}{{s}^{3}}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}}{{s}^{3}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{3} \cdot \frac{-1}{6}} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}}{{s}^{3}}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      17. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      19. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      20. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      21. *-lowering-*.f32 89.4

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   8. Simplified 89.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{s \cdot \left(s \cdot s\right)}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot s\right) \cdot s}}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right)} \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \left(s \cdot s\right) \cdot \color{blue}{\frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{\mathsf{fma}\left(s, \frac{1}{2} \cdot \left(x \cdot x\right), x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2}}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2}}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      12. associate-*l* N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{6}\right)}\right)} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{6}\right)}\right)} \]

      15. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

      16. *-lowering-*.f32 85.0

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot 0.5, \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right)} \]

   10. Applied egg-rr 85.0%

       \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot 0.5, \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}} \]

   11. Taylor expanded in s around 0

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

       1. *-commutative N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{{x}^{3} \cdot \frac{-1}{6}}} \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{{x}^{3} \cdot \frac{-1}{6}}} \]

       3. cube-mult N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6}} \]

       4. unpow2 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6}} \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{\left(x \cdot {x}^{2}\right)} \cdot \frac{-1}{6}} \]

       6. unpow2 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{-1}{6}} \]

       7. *-lowering-*.f32 85.0

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.16666666666666666} \]

   13. Simplified 85.0%

       \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{s}{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 10000.0) 0.5 (/ (* 2.0 (* s s)) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 10000.0f) {
		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) <= 10000.0e0) 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(10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(2.0) * 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(10000.0))
		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) < 1e4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.2%

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

   if 1e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. associate-*r/ N/A

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

      14. distribute-rgt-out N/A

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

      15. accelerator-lowering-fma.f32 N/A

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

   5. Simplified 78.7%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f32 N/A

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

      3. *-lowering-*.f32 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f32 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f32 82.8

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

   8. Simplified 82.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(s \cdot s\right)}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 10000.0) 0.5 (* 2.0 (* s (/ s (* x x))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 10000.0f) {
		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) <= 10000.0e0) 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(10000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(2.0) * Float32(s * Float32(s / Float32(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(10000.0))
		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) < 1e4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 50.2%

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

   if 1e4 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]
   4. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}}{s}}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{2 + \color{blue}{\frac{-1 \cdot \left(x + -1 \cdot \frac{\frac{-1}{6} \cdot \frac{{x}^{3}}{s} + \frac{1}{2} \cdot {x}^{2}}{s}\right)}{s}}} \]

   5. Simplified 86.4%

      \[\leadsto \frac{1}{\color{blue}{2 + \frac{\frac{\mathsf{fma}\left(\frac{x \cdot \left(x \cdot x\right)}{s}, -0.16666666666666666, 0.5 \cdot \left(x \cdot x\right)\right)}{s} - x}{s}}} \]

   6. Taylor expanded in s around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot s\right)}}{{s}^{3}}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{\frac{-1}{6} \cdot {x}^{3} + \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}}{{s}^{3}}} \]

      3. /-lowering-/.f32 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-1}{6} \cdot {x}^{3} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{3} \cdot \frac{-1}{6}} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      5. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      6. unpow2 N/A

         \[\leadsto \frac{1}{\frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot \frac{-1}{6} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      7. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left({x}^{2} \cdot \frac{-1}{6}\right)} + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s}{{s}^{3}}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}}{{s}^{3}}} \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      10. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot s\right)}{{s}^{3}}} \]

      12. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      13. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, \color{blue}{s \cdot \left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)}\right)}{{s}^{3}}} \]

      15. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{{s}^{3}}} \]

      17. cube-mult N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot \left(s \cdot s\right)}}} \]

      18. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{{s}^{2}}}} \]

      19. *-lowering-*.f32 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{s \cdot {s}^{2}}}} \]

      20. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \frac{-1}{6}, s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

      21. *-lowering-*.f32 89.4

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}} \]

   8. Simplified 89.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, s \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)}{s \cdot \left(s \cdot s\right)}}} \]
   9. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{s \cdot \left(s \cdot s\right)}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(s \cdot s\right) \cdot s}}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \color{blue}{\left(s \cdot s\right)} \cdot \frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)} \]

      6. /-lowering-/.f32 N/A

         \[\leadsto \left(s \cdot s\right) \cdot \color{blue}{\frac{s}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) + s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right)}} \]

      7. +-commutative N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{s \cdot \left(\frac{1}{2} \cdot \left(x \cdot x\right)\right) + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)}} \]

      8. accelerator-lowering-fma.f32 N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\color{blue}{\mathsf{fma}\left(s, \frac{1}{2} \cdot \left(x \cdot x\right), x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)}} \]

      9. *-commutative N/A

         \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2}}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      10. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right) \cdot \frac{1}{2}}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{2}, x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)} \]

      12. associate-*l* N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, x \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)}\right)} \]

      13. associate-*r* N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{6}\right)}\right)} \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot \frac{-1}{6}\right)}\right)} \]

      15. *-lowering-*.f32 N/A

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot \frac{1}{2}, \color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \frac{-1}{6}\right)\right)} \]

      16. *-lowering-*.f32 85.0

          \[\leadsto \left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot 0.5, \left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot -0.16666666666666666\right)}\right)} \]

   10. Applied egg-rr 85.0%

       \[\leadsto \color{blue}{\left(s \cdot s\right) \cdot \frac{s}{\mathsf{fma}\left(s, \left(x \cdot x\right) \cdot 0.5, \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\right)}} \]

   11. Taylor expanded in s around inf

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

       1. *-lowering-*.f32 N/A

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

       4. *-lowering-*.f32 N/A

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

       5. /-lowering-/.f32 N/A

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

       6. unpow2 N/A

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

       7. *-lowering-*.f32 77.2

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

   13. Simplified 77.2%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 10000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq -4:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) -4.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= -4.0f) {
		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) <= (-4.0e0)) 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(-4.0))
		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(-4.0))
		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) < -4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.1%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 63.8

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

   5. Simplified 63.8%

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

4. Final simplification 51.8%

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

Alternative 17: 47.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.4%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

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

      4. neg-lowering-neg.f32 43.2

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

   8. Simplified 43.2%

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

      1. clear-num N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-frac N/A

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

      4. frac-2neg N/A

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

      5. /-lowering-/.f32 N/A

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

      6. /-lowering-/.f32 46.9

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

   10. Applied egg-rr 46.9%

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

4. Final simplification 50.1%

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

Alternative 18: 46.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.4%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

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

      4. neg-lowering-neg.f32 43.2

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

   8. Simplified 43.2%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. frac-2neg N/A

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

      6. /-lowering-/.f32 43.2

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

   10. Applied egg-rr 43.2%

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

4. Final simplification 48.5%

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

Alternative 19: 46.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 52.4%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f32 N/A

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

      4. /-lowering-/.f32 46.9

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

   5. Simplified 46.9%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f32 N/A

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

      4. neg-lowering-neg.f32 43.2

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

   8. Simplified 43.2%

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

4. Final simplification 48.5%

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

Alternative 20: 35.5% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 33.2%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))