Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 17

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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) * (-0.3333333333333333e0))) / exp((0.6666666666666666e0 / (s / x)))))
end function

Julia

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

MATLAB

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(e^{1 \cdot \frac{\mathsf{neg}\left(x\right)}{s}}\right)\right)\right) \]

   2. exp-prod N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(e^{1}\right)}^{\color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\left(e^{1}\right), \color{blue}{\left(\frac{\mathsf{neg}\left(x\right)}{s}\right)}\right)\right)\right) \]

   4. exp-1-e N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{s}\right)\right)\right)\right) \]

   6. distribute-frac-neg N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \left(0 - \color{blue}{\frac{x}{s}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{x}{s}\right)}\right)\right)\right)\right) \]

   9. /-lowering-/.f32 99.7%

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{pow.f32}\left(\mathsf{E.f32}\left(\right), \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{1}{1 + \color{blue}{{e}^{\left(0 - \frac{x}{s}\right)}}} \]
5. Step-by-step derivation

   1. add-cube-cbrt N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(\left(\sqrt[3]{\mathsf{E}\left(\right)} \cdot \sqrt[3]{\mathsf{E}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(\color{blue}{0} - \frac{x}{s}\right)}\right)\right)\right) \]

   2. unpow-prod-down N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left({\left(\sqrt[3]{\mathsf{E}\left(\right)} \cdot \sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(0 - \frac{x}{s}\right)} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(0 - \frac{x}{s}\right)}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left({\left(\sqrt[3]{\mathsf{E}\left(\right)} \cdot \sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(0 - \frac{x}{s}\right)}\right), \color{blue}{\left({\left(\sqrt[3]{\mathsf{E}\left(\right)}\right)}^{\left(0 - \frac{x}{s}\right)}\right)}\right)\right)\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{0.6666666666666666 \cdot \frac{x}{s}}} \cdot \frac{1}{e^{0.3333333333333333 \cdot \frac{x}{s}}}}} \]
7. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(\frac{1 \cdot \frac{1}{e^{\frac{1}{3} \cdot \frac{x}{s}}}}{\color{blue}{e^{\frac{2}{3} \cdot \frac{x}{s}}}}\right)\right)\right) \]

   2. div-inv N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \left(\frac{\frac{1}{e^{\frac{1}{3} \cdot \frac{x}{s}}}}{e^{\color{blue}{\frac{2}{3} \cdot \frac{x}{s}}}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{1}{e^{\frac{1}{3} \cdot \frac{x}{s}}}\right), \color{blue}{\left(e^{\frac{2}{3} \cdot \frac{x}{s}}\right)}\right)\right)\right) \]

   4. rec-exp N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\left(e^{\mathsf{neg}\left(\frac{1}{3} \cdot \frac{x}{s}\right)}\right), \left(e^{\color{blue}{\frac{2}{3} \cdot \frac{x}{s}}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{x}{s}\right)\right)\right), \left(e^{\color{blue}{\frac{2}{3} \cdot \frac{x}{s}}}\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\mathsf{neg}\left(\frac{x}{s} \cdot \frac{1}{3}\right)\right)\right), \left(e^{\frac{2}{3} \cdot \frac{x}{s}}\right)\right)\right)\right) \]

   7. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\left(\frac{x}{s} \cdot \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(e^{\color{blue}{\frac{2}{3}} \cdot \frac{x}{s}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\left(\frac{x}{s}\right), \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(e^{\color{blue}{\frac{2}{3}} \cdot \frac{x}{s}}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \left(e^{\frac{2}{3} \cdot \frac{x}{s}}\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \left(e^{\frac{2}{3} \cdot \frac{x}{s}}\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \mathsf{exp.f32}\left(\left(\frac{2}{3} \cdot \frac{x}{s}\right)\right)\right)\right)\right) \]

   12. clear-num N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \mathsf{exp.f32}\left(\left(\frac{2}{3} \cdot \frac{1}{\frac{s}{x}}\right)\right)\right)\right)\right) \]

   13. un-div-inv N/A

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \mathsf{exp.f32}\left(\left(\frac{\frac{2}{3}}{\frac{s}{x}}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\frac{2}{3}, \left(\frac{s}{x}\right)\right)\right)\right)\right)\right) \]

   15. /-lowering-/.f32 99.8%

       \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{/.f32}\left(\mathsf{exp.f32}\left(\mathsf{*.f32}\left(\mathsf{/.f32}\left(x, s\right), \frac{-1}{3}\right)\right), \mathsf{exp.f32}\left(\mathsf{/.f32}\left(\frac{2}{3}, \mathsf{/.f32}\left(s, x\right)\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{0 - \frac{x}{s}}} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (+ 1.0 (exp (- 0.0 (/ x s))))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + expf((0.0f - (x / s))));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + exp((0.0e0 - (x / s))))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + exp(Float32(Float32(0.0) - Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + exp((single(0.0) - (x / s))));
end

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + e^{0 - \frac{x}{s}}} \]
4. Add Preprocessing

Alternative 3: 65.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(x \cdot \left(\frac{x \cdot \frac{-0.16666666666666666}{s \cdot s}}{s} + \frac{0.5}{s \cdot s}\right) + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999593223797e-31)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* x (+ (/ (* x (/ -0.16666666666666666 (* s s))) s) (/ 0.5 (* s s))))
       (/ -1.0 s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f + (x * ((x * (((x * (-0.16666666666666666f / (s * s))) / s) + (0.5f / (s * s)))) + (-1.0f / s))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999593223797e-31)) then
        tmp = 1.0e0 / (2.0e0 + (x * ((x * (((x * ((-0.16666666666666666e0) / (s * s))) / s) + (0.5e0 / (s * s)))) + ((-1.0e0) / s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(x * Float32(Float32(Float32(x * Float32(Float32(-0.16666666666666666) / Float32(s * s))) / s) + Float32(Float32(0.5) / Float32(s * s)))) + Float32(Float32(-1.0) / s)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) + (x * ((x * (((x * (single(-0.16666666666666666) / (s * s))) / s) + (single(0.5) / (s * s)))) + (single(-1.0) / s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.8%

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

      1. times-frac N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{x}{s} \cdot \frac{\frac{-1}{6}}{s \cdot s}\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\left(\frac{x \cdot \frac{\frac{-1}{6}}{s \cdot s}}{s}\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\left(x \cdot \frac{\frac{-1}{6}}{s \cdot s}\right), s\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6}}{s \cdot s}\right)\right), s\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \left(s \cdot s\right)\right)\right), s\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 88.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(s, s\right)\right)\right), s\right), \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 88.6%

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

   if -1.99999996e-31 < x

   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 48.1%

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

Alternative 4: 65.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999593223797e-31)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (/ -1.0 s)
       (* x (/ (+ 0.5 (/ (* x -0.16666666666666666) s)) (* s s)))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((0.5f + ((x * -0.16666666666666666f) / s)) / (s * s))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999593223797e-31)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((0.5e0 + ((x * (-0.16666666666666666e0)) / s)) / (s * s))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(Float32(0.5) + Float32(Float32(x * Float32(-0.16666666666666666)) / s)) / Float32(s * s)))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((single(0.5) + ((x * single(-0.16666666666666666)) / s)) / (s * s))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.8%

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

   6. Taylor expanded in s around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{1}{2} + \frac{-1}{6} \cdot \frac{x}{s}\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot \frac{x}{s}\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \left(\frac{\frac{-1}{6} \cdot x}{s}\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f32 88.6%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{+.f32}\left(\frac{1}{2}, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), s\right)\right), \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 88.6%

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

   if -1.99999996e-31 < x

   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 48.1%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5 + \frac{x \cdot -0.16666666666666666}{s}}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{x \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999593223797e-31)
   (/
    1.0
    (+
     2.0
     (* x (+ (/ -1.0 s) (* x (/ (* x -0.16666666666666666) (* s (* s s))))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * ((x * -0.16666666666666666f) / (s * (s * s)))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999593223797e-31)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * ((x * (-0.16666666666666666e0)) / (s * (s * s)))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(x * Float32(-0.16666666666666666)) / Float32(s * Float32(s * s))))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * ((x * single(-0.16666666666666666)) / (s * (s * s)))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6} \cdot x}{{s}^{3}}\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot x\right), \left({s}^{3}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(x \cdot \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot \left(s \cdot s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \left(s \cdot {s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \left({s}^{2}\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \left(s \cdot s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f32 86.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, s\right)\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 86.2%

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

   if -1.99999996e-31 < x

   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 48.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{x \cdot -0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-0.16666666666666666 \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.99999991225835e-15)
   (/ 1.0 (+ 2.0 (* x (/ (* -0.16666666666666666 (* x x)) (* s (* s s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.99999991225835e-15f) {
		tmp = 1.0f / (2.0f + (x * ((-0.16666666666666666f * (x * x)) / (s * (s * s)))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.99999991225835e-15)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-0.16666666666666666e0) * (x * x)) / (s * (s * s)))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.99999991225835e-15))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-0.16666666666666666) * Float32(x * x)) / Float32(s * Float32(s * s))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.99999991225835e-15))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-0.16666666666666666) * (x * x)) / (s * (s * s)))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999991e-15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{2}}{{s}^{3}}\right)}\right)\right)\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \left(\frac{\frac{-1}{6} \cdot {x}^{2}}{\color{blue}{{s}^{3}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot {x}^{2}\right), \color{blue}{\left({s}^{3}\right)}\right)\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\left({x}^{2} \cdot \frac{-1}{6}\right), \left({\color{blue}{s}}^{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left({x}^{2}\right), \frac{-1}{6}\right), \left({\color{blue}{s}}^{3}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\left(x \cdot x\right), \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \left({s}^{3}\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \left(s \cdot \color{blue}{\left(s \cdot s\right)}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \left(s \cdot {s}^{\color{blue}{2}}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \color{blue}{\left({s}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \left(s \cdot \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 86.3%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \frac{-1}{6}\right), \mathsf{*.f32}\left(s, \mathsf{*.f32}\left(s, \color{blue}{s}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 86.3%

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

   if -4.99999991e-15 < x

   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 49.9%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{-0.16666666666666666 \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999593223797e-31)
   (/ 1.0 (+ 2.0 (* x (+ (/ -1.0 s) (* x (/ 0.5 (* s s)))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f + (x * ((-1.0f / s) + (x * (0.5f / (s * s))))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999593223797e-31)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((-1.0e0) / s) + (x * (0.5e0 / (s * s))))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(-1.0) / s) + Float32(x * Float32(Float32(0.5) / Float32(s * s)))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) + (x * ((single(-1.0) / s) + (x * (single(0.5) / (s * s))))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 86.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \color{blue}{\left(\frac{\frac{1}{2}}{{s}^{2}}\right)}\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left({s}^{2}\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \left(s \cdot s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 80.0%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(x, \mathsf{+.f32}\left(\mathsf{*.f32}\left(x, \mathsf{/.f32}\left(\frac{1}{2}, \mathsf{*.f32}\left(s, s\right)\right)\right), \mathsf{/.f32}\left(-1, s\right)\right)\right)\right)\right) \]

   8. Simplified 80.0%

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

   if -1.99999996e-31 < x

   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 48.1%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{-1}{s} + x \cdot \frac{0.5}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (/ 1.0 (/ (* -0.16666666666666666 (* x (* x x))) (* s (* s s))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = 1.0f / ((-0.16666666666666666f * (x * (x * x))) / (s * (s * s)));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = 1.0e0 / (((-0.16666666666666666e0) * (x * (x * x))) / (s * (s * s)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(-0.16666666666666666) * Float32(x * Float32(x * x))) / Float32(s * Float32(s * s))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = single(1.0) / ((single(-0.16666666666666666) * (x * (x * x))) / (s * (s * s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot 1\right)}{{s}^{3}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {1}^{3}\right)}{{s}^{3}}\right)\right) \]

      4. log-E N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{{s}^{3}}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot {s}^{\color{blue}{2}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s}}{\color{blue}{{s}^{2}}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}}{{\color{blue}{s}}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

   8. Simplified 82.3%

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

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

         \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{\frac{-1}{6} \cdot \frac{x \cdot \left(x \cdot x\right)}{s}}{s \cdot s}\right)}\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s}}{\color{blue}{s} \cdot s}\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{\color{blue}{\left(s \cdot s\right) \cdot s}}\right)\right) \]

      4. pow3 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)}{{s}^{\color{blue}{3}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), \color{blue}{\left({s}^{3}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \left(x \cdot \left(x \cdot x\right)\right)\right), \left({\color{blue}{s}}^{3}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \left(x \cdot x\right)\right)\right), \left({s}^{3}\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \left({s}^{3}\right)\right)\right) \]

      9. pow3 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \left(\left(s \cdot s\right) \cdot \color{blue}{s}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(\left(s \cdot s\right), \color{blue}{s}\right)\right)\right) \]

      11. *-lowering-*.f32 84.2%

          \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\mathsf{*.f32}\left(\frac{-1}{6}, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), s\right)\right)\right) \]

   10. Applied egg-rr 84.2%

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

   if -1.99999996e-13 < x

   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 50.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{-0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-6}{\frac{x \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (/ -6.0 (/ (* x (* x x)) (* s (* s s))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = -6.0f / ((x * (x * x)) / (s * (s * s)));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = (-6.0e0) / ((x * (x * x)) / (s * (s * s)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(-6.0) / Float32(Float32(x * Float32(x * x)) / Float32(s * Float32(s * s))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = single(-6.0) / ((x * (x * x)) / (s * (s * s)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot 1\right)}{{s}^{3}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {1}^{3}\right)}{{s}^{3}}\right)\right) \]

      4. log-E N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{{s}^{3}}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot {s}^{\color{blue}{2}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s}}{\color{blue}{{s}^{2}}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}}{{\color{blue}{s}}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

   8. Simplified 82.3%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(\left({\frac{-1}{6}}^{-1}\right), \color{blue}{\left(\frac{\frac{x \cdot \left(x \cdot x\right)}{s}}{s \cdot s}\right)}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(-6, \left(\frac{\color{blue}{\frac{x \cdot \left(x \cdot x\right)}{s}}}{s \cdot s}\right)\right) \]

      7. associate-/l/ N/A

         \[\leadsto \mathsf{/.f32}\left(-6, \left(\frac{x \cdot \left(x \cdot x\right)}{\color{blue}{\left(s \cdot s\right) \cdot s}}\right)\right) \]

      8. pow3 N/A

         \[\leadsto \mathsf{/.f32}\left(-6, \left(\frac{x \cdot \left(x \cdot x\right)}{{s}^{\color{blue}{3}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), \color{blue}{\left({s}^{3}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), \left({\color{blue}{s}}^{3}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \left({s}^{3}\right)\right)\right) \]

      12. pow3 N/A

          \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \left(\left(s \cdot s\right) \cdot \color{blue}{s}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{*.f32}\left(\left(s \cdot s\right), \color{blue}{s}\right)\right)\right) \]

      14. *-lowering-*.f32 84.2%

          \[\leadsto \mathsf{/.f32}\left(-6, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(s, s\right), s\right)\right)\right) \]

   10. Applied egg-rr 84.2%

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

   if -1.99999996e-13 < x

   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 50.1%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-6}{\frac{x \cdot \left(x \cdot x\right)}{s \cdot \left(s \cdot s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{s}{\frac{x \cdot \left(x \cdot x\right)}{s}} \cdot \frac{s}{-0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (* (/ s (/ (* x (* x x)) s)) (/ s -0.16666666666666666))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = (s / ((x * (x * x)) / s)) * (s / -0.16666666666666666f);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = (s / ((x * (x * x)) / s)) * (s / (-0.16666666666666666e0))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(s / Float32(Float32(x * Float32(x * x)) / s)) * Float32(s / Float32(-0.16666666666666666)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = (s / ((x * (x * x)) / s)) * (s / single(-0.16666666666666666));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot 1\right)}{{s}^{3}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {1}^{3}\right)}{{s}^{3}}\right)\right) \]

      4. log-E N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{{s}^{3}}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot {s}^{\color{blue}{2}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s}}{\color{blue}{{s}^{2}}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}}{{\color{blue}{s}}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

   8. Simplified 82.3%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{s}{\frac{x \cdot \left(x \cdot x\right)}{s}}\right), \color{blue}{\left(\frac{s}{\frac{-1}{6}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \left(\frac{x \cdot \left(x \cdot x\right)}{s}\right)\right), \left(\frac{\color{blue}{s}}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

      9. /-lowering-/.f32 79.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), s\right)\right), \mathsf{/.f32}\left(s, \color{blue}{\frac{-1}{6}}\right)\right) \]

   10. Applied egg-rr 79.8%

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

   if -1.99999996e-13 < x

   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 50.1%

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

Alternative 11: 59.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{s}{-0.16666666666666666} \cdot \frac{s}{\frac{x}{s} \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (* (/ s -0.16666666666666666) (/ s (* (/ x s) (* x x))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = (s / -0.16666666666666666f) * (s / ((x / s) * (x * x)));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = (s / (-0.16666666666666666e0)) * (s / ((x / s) * (x * x)))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(s / Float32(-0.16666666666666666)) * Float32(s / Float32(Float32(x / s) * Float32(x * x))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = (s / single(-0.16666666666666666)) * (s / ((x / s) * (x * x)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot 1\right)}{{s}^{3}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {1}^{3}\right)}{{s}^{3}}\right)\right) \]

      4. log-E N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{{s}^{3}}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot {s}^{\color{blue}{2}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s}}{\color{blue}{{s}^{2}}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}}{{\color{blue}{s}}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

   8. Simplified 82.3%

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

      1. clear-num N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f32}\left(\left(\frac{s}{\frac{x \cdot \left(x \cdot x\right)}{s}}\right), \color{blue}{\left(\frac{s}{\frac{-1}{6}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \left(\frac{x \cdot \left(x \cdot x\right)}{s}\right)\right), \left(\frac{\color{blue}{s}}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\left(x \cdot \left(x \cdot x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \left(x \cdot x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), s\right)\right), \left(\frac{s}{\frac{-1}{6}}\right)\right) \]

      9. /-lowering-/.f32 79.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, x\right)\right), s\right)\right), \mathsf{/.f32}\left(s, \color{blue}{\frac{-1}{6}}\right)\right) \]

   10. Applied egg-rr 79.8%

       \[\leadsto \color{blue}{\frac{s}{\frac{x \cdot \left(x \cdot x\right)}{s}} \cdot \frac{s}{-0.16666666666666666}} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \left(\frac{\left(x \cdot x\right) \cdot x}{s}\right)\right), \mathsf{/.f32}\left(s, \frac{-1}{6}\right)\right) \]

       2. associate-/l* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \left(\left(x \cdot x\right) \cdot \frac{x}{s}\right)\right), \mathsf{/.f32}\left(s, \frac{-1}{6}\right)\right) \]

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

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{*.f32}\left(\left(x \cdot x\right), \left(\frac{x}{s}\right)\right)\right), \mathsf{/.f32}\left(s, \frac{-1}{6}\right)\right) \]

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

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \left(\frac{x}{s}\right)\right)\right), \mathsf{/.f32}\left(s, \frac{-1}{6}\right)\right) \]

       5. /-lowering-/.f32 79.8%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{/.f32}\left(s, \mathsf{*.f32}\left(\mathsf{*.f32}\left(x, x\right), \mathsf{/.f32}\left(x, s\right)\right)\right), \mathsf{/.f32}\left(s, \frac{-1}{6}\right)\right) \]

   12. Applied egg-rr 79.8%

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

   if -1.99999996e-13 < x

   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 50.1%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{s}{-0.16666666666666666} \cdot \frac{s}{\frac{x}{s} \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;s \cdot \frac{s}{\frac{-0.16666666666666666}{\frac{s}{x \cdot \left(x \cdot x\right)}}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (* s (/ s (/ -0.16666666666666666 (/ s (* x (* x x))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = s * (s / (-0.16666666666666666f / (s / (x * (x * x)))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = s * (s / ((-0.16666666666666666e0) / (s / (x * (x * x)))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(s * Float32(s / Float32(Float32(-0.16666666666666666) / Float32(s / Float32(x * Float32(x * x))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = s * (s / (single(-0.16666666666666666) / (s / (x * (x * x)))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)\right)}\right) \]

   5. Simplified 88.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot \frac{{x}^{3}}{{s}^{3}}\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot {x}^{3}}{\color{blue}{{s}^{3}}}\right)\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot 1\right)}{{s}^{3}}\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {1}^{3}\right)}{{s}^{3}}\right)\right) \]

      4. log-E N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{{s}^{3}}\right)\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot \color{blue}{\left(s \cdot s\right)}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s \cdot {s}^{\color{blue}{2}}}\right)\right) \]

      7. associate-/r* N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{\frac{-1}{6} \cdot \left({x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}\right)}{s}}{\color{blue}{{s}^{2}}}\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(\frac{\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}}{{\color{blue}{s}}^{2}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{/.f32}\left(\left(\frac{-1}{6} \cdot \frac{{x}^{3} \cdot {\log \mathsf{E}\left(\right)}^{3}}{s}\right), \color{blue}{\left({s}^{2}\right)}\right)\right) \]

   8. Simplified 82.3%

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

      1. clear-num N/A

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

      2. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f32}\left(s, \color{blue}{\left(\frac{s}{\frac{-1}{6} \cdot \frac{x \cdot \left(x \cdot x\right)}{s}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \color{blue}{\left(\frac{-1}{6} \cdot \frac{x \cdot \left(x \cdot x\right)}{s}\right)}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \left(\frac{-1}{6} \cdot \frac{1}{\color{blue}{\frac{s}{x \cdot \left(x \cdot x\right)}}}\right)\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \left(\frac{\frac{-1}{6}}{\color{blue}{\frac{s}{x \cdot \left(x \cdot x\right)}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\frac{-1}{6}, \color{blue}{\left(\frac{s}{x \cdot \left(x \cdot x\right)}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(s, \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \color{blue}{\left(x \cdot x\right)}\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f32 79.8%

          \[\leadsto \mathsf{*.f32}\left(s, \mathsf{/.f32}\left(s, \mathsf{/.f32}\left(\frac{-1}{6}, \mathsf{/.f32}\left(s, \mathsf{*.f32}\left(x, \mathsf{*.f32}\left(x, \color{blue}{x}\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 79.8%

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

   if -1.99999996e-13 < x

   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 50.1%

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

Alternative 13: 51.0% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{s \cdot \left(-s\right)}{s}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13) (/ (/ (* s (- s)) s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = ((s * -s) / s) / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = ((s * -s) / s) / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(Float32(s * Float32(-s)) / s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = ((s * -s) / s) / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 48.9%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 48.9%

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

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

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

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 43.1%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 43.1%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-neg-frac N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(s\right)\right), \color{blue}{x}\right) \]

      4. neg-lowering-neg.f32 43.1%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(s\right), x\right) \]

   10. Applied egg-rr 43.1%

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

       1. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(0 - s\right), x\right) \]

       2. flip-- N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\frac{0 \cdot 0 - s \cdot s}{0 + s}\right), x\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\frac{0 - s \cdot s}{0 + s}\right), x\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\left(\frac{\mathsf{neg}\left(s \cdot s\right)}{0 + s}\right), x\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(\mathsf{neg}\left(s \cdot s\right)\right), \left(0 + s\right)\right), x\right) \]

       6. neg-sub0 N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\left(0 - s \cdot s\right), \left(0 + s\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \left(s \cdot s\right)\right), \left(0 + s\right)\right), x\right) \]

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(s, s\right)\right), \left(0 + s\right)\right), x\right) \]

       9. +-lowering-+.f32 58.9%

          \[\leadsto \mathsf{/.f32}\left(\mathsf{/.f32}\left(\mathsf{\_.f32}\left(0, \mathsf{*.f32}\left(s, s\right)\right), \mathsf{+.f32}\left(0, s\right)\right), x\right) \]

   12. Applied egg-rr 58.9%

       \[\leadsto \frac{\color{blue}{\frac{0 - s \cdot s}{0 + s}}}{x} \]

   if -1.99999996e-13 < x

   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 50.1%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{s \cdot \left(-s\right)}{s}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999593223797 \cdot 10^{-31}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999593223797e-31) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999593223797e-31f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999593223797e-31)) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999593223797e-31))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999593223797e-31))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-31

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 52.2%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 52.2%

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

   if -1.99999996e-31 < x

   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 48.1%

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

Alternative 15: 47.5% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9) (/ -1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = -1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = (-1.0e0) / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = single(-1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999997e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 52.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 52.3%

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

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

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

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 48.2%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. clear-num N/A

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

      3. distribute-neg-frac N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(-1, \color{blue}{\left(\frac{x}{s}\right)}\right) \]

      6. /-lowering-/.f32 52.3%

         \[\leadsto \mathsf{/.f32}\left(-1, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right) \]

   10. Applied egg-rr 52.3%

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

   if -4.99999997e-9 < x

   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 48.2%

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

Alternative 16: 46.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4.999999969612645e-9) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4.999999969612645e-9f) {
		tmp = s / -x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4.999999969612645e-9)) then
        tmp = s / -x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4.999999969612645e-9))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4.999999969612645e-9))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -4.99999997e-9

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f32}\left(1, \color{blue}{\left(2 + -1 \cdot \frac{x}{s}\right)}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 + \left(\mathsf{neg}\left(\frac{x}{s}\right)\right)\right)\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{/.f32}\left(1, \left(2 - \color{blue}{\frac{x}{s}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \color{blue}{\left(\frac{x}{s}\right)}\right)\right) \]

      4. /-lowering-/.f32 52.3%

         \[\leadsto \mathsf{/.f32}\left(1, \mathsf{\_.f32}\left(2, \mathsf{/.f32}\left(x, \color{blue}{s}\right)\right)\right) \]

   5. Simplified 52.3%

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

   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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{s}{x}} \]

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

         \[\leadsto \mathsf{\_.f32}\left(0, \color{blue}{\left(\frac{s}{x}\right)}\right) \]

      4. /-lowering-/.f32 48.2%

         \[\leadsto \mathsf{\_.f32}\left(0, \mathsf{/.f32}\left(s, \color{blue}{x}\right)\right) \]

   8. Simplified 48.2%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. distribute-neg-frac N/A

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

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

         \[\leadsto \mathsf{/.f32}\left(\left(\mathsf{neg}\left(s\right)\right), \color{blue}{x}\right) \]

      4. neg-lowering-neg.f32 48.2%

         \[\leadsto \mathsf{/.f32}\left(\mathsf{neg.f32}\left(s\right), x\right) \]

   10. Applied egg-rr 48.2%

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

   if -4.99999997e-9 < x

   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 48.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.999999969612645 \cdot 10^{-9}:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.9% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 35.7%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))