Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(e^{-2}\right)}^{\left(\mathsf{log1p}\left(e^{-\frac{x}{s}}\right) \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (pow (exp -2.0) (* (log1p (exp (- (/ x s)))) 0.5)))

C

float code(float x, float s) {
	return powf(expf(-2.0f), (log1pf(expf(-(x / s))) * 0.5f));
}

Julia

function code(x, s)
	return exp(Float32(-2.0)) ^ Float32(log1p(exp(Float32(-Float32(x / s)))) * Float32(0.5))
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. sqr-pow N/A

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

   3. pow2 N/A

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

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

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

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

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

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

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

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

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

   8. distribute-frac-neg N/A

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

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

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

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

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

   11. metadata-eval 99.3

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

4. Applied egg-rr 99.3%

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

   1. pow-to-exp N/A

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

   2. pow-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

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

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

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

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

   7. *-commutative N/A

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

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

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

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

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

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

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

   11. distribute-neg-frac2 N/A

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

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

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

   13. neg-lowering-neg.f32 99.9

       \[\leadsto {\left(e^{2}\right)}^{\left(-0.5 \cdot \mathsf{log1p}\left(e^{\frac{x}{\color{blue}{-s}}}\right)\right)} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left(e^{2}\right)}^{\left(-0.5 \cdot \mathsf{log1p}\left(e^{\frac{x}{-s}}\right)\right)}} \]
7. Step-by-step derivation

   1. sqr-pow N/A

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

   2. pow-prod-down N/A

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

   3. associate-/l* N/A

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

   4. pow-unpow N/A

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

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

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

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

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

   7. prod-exp N/A

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

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

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

   9. metadata-eval N/A

      \[\leadsto {\left({\left(e^{\color{blue}{4}}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

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

       \[\leadsto {\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}^{\color{blue}{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)}} \]

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

       \[\leadsto {\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{\color{blue}{\mathsf{log1p}\left(e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}}{2}\right)} \]

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

       \[\leadsto {\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{\mathsf{log1p}\left(\color{blue}{e^{\frac{x}{\mathsf{neg}\left(s\right)}}}\right)}{2}\right)} \]

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

       \[\leadsto {\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{\mathsf{log1p}\left(e^{\color{blue}{\frac{x}{\mathsf{neg}\left(s\right)}}}\right)}{2}\right)} \]

   14. neg-lowering-neg.f32 99.9

       \[\leadsto {\left({\left(e^{4}\right)}^{-0.5}\right)}^{\left(\frac{\mathsf{log1p}\left(e^{\frac{x}{\color{blue}{-s}}}\right)}{2}\right)} \]

8. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{{\left({\left(e^{4}\right)}^{-0.5}\right)}^{\left(\frac{\mathsf{log1p}\left(e^{\frac{x}{-s}}\right)}{2}\right)}} \]
9. Step-by-step derivation

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

      \[\leadsto \color{blue}{{\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)}} \]

   2. pow-exp N/A

      \[\leadsto {\color{blue}{\left(e^{4 \cdot \frac{-1}{2}}\right)}}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

   3. rem-log-exp N/A

      \[\leadsto {\left(e^{\color{blue}{\log \left(e^{4 \cdot \frac{-1}{2}}\right)}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

   4. pow-exp N/A

      \[\leadsto {\left(e^{\log \color{blue}{\left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

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

      \[\leadsto {\color{blue}{\left(e^{\log \left({\left(e^{4}\right)}^{\frac{-1}{2}}\right)}\right)}}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

   6. pow-exp N/A

      \[\leadsto {\left(e^{\log \color{blue}{\left(e^{4 \cdot \frac{-1}{2}}\right)}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

   7. rem-log-exp N/A

      \[\leadsto {\left(e^{\color{blue}{4 \cdot \frac{-1}{2}}}\right)}^{\left(\frac{\log \left(1 + e^{\frac{x}{\mathsf{neg}\left(s\right)}}\right)}{2}\right)} \]

   8. metadata-eval N/A

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

   9. div-inv N/A

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

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

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

   15. neg-lowering-neg.f32 99.9

       \[\leadsto {\left(e^{-2}\right)}^{\left(\mathsf{log1p}\left(e^{\frac{x}{\color{blue}{-s}}}\right) \cdot 0.5\right)} \]

10. Applied egg-rr 99.9%

    \[\leadsto \color{blue}{{\left(e^{-2}\right)}^{\left(\mathsf{log1p}\left(e^{\frac{x}{-s}}\right) \cdot 0.5\right)}} \]

11. Final simplification 99.9%

    \[\leadsto {\left(e^{-2}\right)}^{\left(\mathsf{log1p}\left(e^{-\frac{x}{s}}\right) \cdot 0.5\right)} \]
12. Add Preprocessing

Alternative 2: 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 3: 66.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(x, fma(Float32(-0.16666666666666666), Float32(x / Float32(s * Float32(s * s))), Float32(Float32(0.5) / Float32(s * 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) < 1

   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 50.8%

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

   if 1 < (/.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 73.5%

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

   6. Taylor expanded in x around 0

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

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

   8. Simplified 87.6%

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

4. Final simplification 66.7%

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

Alternative 4: 65.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 5.000000136226006 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 5.000000136226006e-28)
   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 <= 5.000000136226006e-28f) {
		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(-x) <= Float32(5.000000136226006e-28))
		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 (neg.f32 x) < 5.00000014e-28

   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 46.3%

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

   if 5.00000014e-28 < (neg.f32 x)

   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 85.8%

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

Alternative 5: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 100.0)
   0.5
   (/ 1.0 (* x (* (* x x) (/ -0.16666666666666666 (* s (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 100.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * ((x * x) * (-0.16666666666666666f / (s * (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) <= 100.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * ((x * x) * ((-0.16666666666666666e0) / (s * (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(100.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(x * x) * Float32(Float32(-0.16666666666666666) / Float32(s * Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(100.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * ((x * x) * (single(-0.16666666666666666) / (s * (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 100

   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 50.3%

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

   if 100 < (/.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 74.5%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 83.1%

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

   9. Taylor expanded in x around inf

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f32 83.1

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

   11. Simplified 83.1%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 100:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   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 50.6%

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

   if 2 < (/.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 73.8%

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

   6. Taylor expanded in x around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. sub-neg N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

   8. Simplified 82.4%

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

   9. Taylor expanded in x around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. *-lowering-*.f32 78.8

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

   11. Simplified 78.8%

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

4. Final simplification 62.7%

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

Alternative 7: 62.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 500.0) 0.5 (/ (* s (* (* s s) -6.0)) (* x (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 500.0f) {
		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) <= 500.0e0) 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(500.0))
		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(500.0))
		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) < 500

   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 49.7%

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

   if 500 < (/.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 75.7%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l/ N/A

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

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

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

      7. associate-*l* N/A

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

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

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

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

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

      10. *-lowering-*.f32 75.7

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

   7. Applied egg-rr 75.7%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-6 \cdot \frac{{s}^{3}}{{x}^{3}}} \]
   9. 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. cube-mult N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 78.1

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

   10. Simplified 78.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 500:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{s \cdot \left(\left(s \cdot s\right) \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- (/ x s)) 50000.0) 0.5 (/ (* (* s s) 2.0) (* x x))))

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 50000.0f) {
		tmp = 0.5f;
	} else {
		tmp = ((s * s) * 2.0f) / (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) <= 50000.0e0) then
        tmp = 0.5e0
    else
        tmp = ((s * s) * 2.0e0) / (x * x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-Float32(x / s)) <= Float32(50000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(Float32(s * s) * Float32(2.0)) / Float32(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(50000.0))
		tmp = single(0.5);
	else
		tmp = ((s * s) * single(2.0)) / (x * x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 5e4

   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 48.6%

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

   if 5e4 < (/.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 73.8%

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

   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 76.1

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

   8. Simplified 76.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\frac{x}{s} \leq 50000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot 2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 28.2%

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

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

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 54.5

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

   5. Simplified 54.5%

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

4. Final simplification 45.4%

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

Alternative 10: 47.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   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 50.6%

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

   if 2 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 34.4

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

   5. Simplified 34.4%

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

   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 30.5

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

   8. Simplified 30.5%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-frac-neg2 N/A

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

      5. frac-2neg N/A

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

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

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

      7. /-lowering-/.f32 34.4

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

   10. Applied egg-rr 34.4%

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

4. Final simplification 43.7%

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

Alternative 11: 46.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (-(x / s) <= 2.0f) {
		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) <= 2.0e0) 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(2.0))
		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(2.0))
		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) < 2

   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 50.6%

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

   if 2 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 34.4

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

   5. Simplified 34.4%

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

   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 30.5

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

   8. Simplified 30.5%

      \[\leadsto \color{blue}{\frac{s}{-x}} \]
   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 30.5

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

   10. Applied egg-rr 30.5%

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

4. Final simplification 42.0%

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

Alternative 12: 46.6% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-(x / s) <= single(2.0))
		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) < 2

   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 50.6%

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

   if 2 < (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + -1 \cdot \frac{x}{s}}} \]
   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 34.4

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

   5. Simplified 34.4%

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

   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 30.5

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

   8. Simplified 30.5%

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

4. Final simplification 42.0%

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

Alternative 13: 35.1% 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 31.6%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))