Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 13.2s

Alternatives: 12

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (Float32(exp(1)) ^ Float32(Float32(0.0) - Float32(x / s)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (single(2.71828182845904523536) ^ (single(0.0) - (x / s))));
end

Derivation

1. Initial program 99.9%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
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. clear-num N/A

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

   7. frac-2neg N/A

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

   8. metadata-eval N/A

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

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

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

   10. distribute-frac-neg2 N/A

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

   11. remove-double-neg N/A

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

   12. /-lowering-/.f32 99.9%

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

4. Applied egg-rr 99.9%

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

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. neg-lowering-neg.f32 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto \frac{1}{1 + {e}^{\left(0 - \frac{x}{s}\right)}} \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (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 + (1.0e0 / exp((x / s))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. distribute-frac-neg N/A

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

   2. exp-neg N/A

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

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

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

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

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

   5. /-lowering-/.f32 99.9%

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

4. Applied egg-rr 99.9%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
5. Add Preprocessing

Alternative 3: 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 99.9%

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

3. Final simplification 99.9%

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

Alternative 4: 65.8% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.000000136226006e-28)
   (/
    1.0
    (+
     2.0
     (*
      x
      (+
       (* (/ x (* s s)) (+ (/ (* x -0.16666666666666666) s) 0.5))
       (/ -1.0 s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.000000136226006e-28f) {
		tmp = 1.0f / (2.0f + (x * (((x / (s * s)) * (((x * -0.16666666666666666f) / s) + 0.5f)) + (-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 <= (-5.000000136226006e-28)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((x / (s * s)) * (((x * (-0.16666666666666666e0)) / s) + 0.5e0)) + ((-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(-5.000000136226006e-28))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(x / Float32(s * s)) * Float32(Float32(Float32(x * Float32(-0.16666666666666666)) / s) + Float32(0.5))) + 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(-5.000000136226006e-28))
		tmp = single(1.0) / (single(2.0) + (x * (((x / (s * s)) * (((x * single(-0.16666666666666666)) / s) + single(0.5))) + (single(-1.0) / s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000014e-28

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 89.8%

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

   if -5.00000014e-28 < 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 47.3%

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

Alternative 5: 61.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{0.5}{x \cdot \left(s \cdot s\right)} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot s\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000036274937e-15)
   (/
    1.0
    (*
     (* x (* x x))
     (+ (/ 0.5 (* x (* s s))) (/ -0.16666666666666666 (* s (* s s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000036274937e-15f) {
		tmp = 1.0f / ((x * (x * x)) * ((0.5f / (x * (s * s))) + (-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 <= (-1.0000000036274937e-15)) then
        tmp = 1.0e0 / ((x * (x * x)) * ((0.5e0 / (x * (s * s))) + ((-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(-1.0000000036274937e-15))
		tmp = Float32(Float32(1.0) / Float32(Float32(x * Float32(x * x)) * Float32(Float32(Float32(0.5) / Float32(x * Float32(s * s))) + Float32(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(-1.0000000036274937e-15))
		tmp = single(1.0) / ((x * (x * x)) * ((single(0.5) / (x * (s * s))) + (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 < -1e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. sub-neg N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

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

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

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

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

      21. cube-mult N/A

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

      22. unpow2 N/A

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

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

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

      24. unpow2 N/A

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

      25. *-lowering-*.f32 93.3%

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

   8. Simplified 93.3%

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

   if -1e-15 < 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 48.5%

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

Alternative 6: 61.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(s \cdot s\right) \cdot \left(s \cdot -6\right)}{x \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000036274937e-15)
   (/ (* (* s s) (* s -6.0)) (* x (* x x)))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000036274937e-15f) {
		tmp = ((s * s) * (s * -6.0f)) / (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 <= (-1.0000000036274937e-15)) then
        tmp = ((s * s) * (s * (-6.0e0))) / (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(-1.0000000036274937e-15))
		tmp = Float32(Float32(Float32(s * s) * Float32(s * Float32(-6.0))) / 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(-1.0000000036274937e-15))
		tmp = ((s * s) * (s * single(-6.0))) / (x * (x * x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-15

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

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

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

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

      2. associate-*r/ N/A

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

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

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow3 N/A

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

      5. unpow2 N/A

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

      6. associate-*l* N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. cube-mult N/A

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

      12. unpow2 N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f32 91.9%

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

   8. Simplified 91.9%

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

   if -1e-15 < 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 48.5%

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

Alternative 7: 57.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\left(s \cdot s\right) \cdot -2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.99999983775159e-18) (/ (/ (* (* s s) -2.0) x) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.99999983775159e-18f) {
		tmp = (((s * s) * -2.0f) / 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 <= (-9.99999983775159e-18)) then
        tmp = (((s * s) * (-2.0e0)) / x) / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.99999983775159e-18))
		tmp = Float32(Float32(Float32(Float32(s * s) * Float32(-2.0)) / 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(-9.99999983775159e-18))
		tmp = (((s * s) * single(-2.0)) / x) / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.99999984e-18

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 41.4%

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

   5. Simplified 41.4%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

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

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-2 \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({s}^{2} \cdot -2\right), x\right), s\right), x\right) \]

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 35.1%

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

   8. Simplified 35.1%

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

   9. Taylor expanded in s around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

       5. *-lowering-*.f32 76.2%

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

   11. Simplified 76.2%

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

   if -9.99999984e-18 < 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 49.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.99999983775159 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\left(s \cdot s\right) \cdot -2}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;-2 \cdot \left(s \cdot \frac{s}{x \cdot x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000036274937e-15) (* -2.0 (* s (/ s (* x x)))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000036274937e-15f) {
		tmp = -2.0f * (s * (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 <= (-1.0000000036274937e-15)) then
        tmp = (-2.0e0) * (s * (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(-1.0000000036274937e-15))
		tmp = Float32(Float32(-2.0) * Float32(s * Float32(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(-1.0000000036274937e-15))
		tmp = single(-2.0) * (s * (s / (x * x)));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-15

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 42.2%

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

   5. Simplified 42.2%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv N/A

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

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

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

      5. cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

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

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(\left(\mathsf{neg}\left(2\right)\right) \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(-2 \cdot {s}^{2}\right), x\right), s\right), x\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{/.f32}\left(\mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left({s}^{2} \cdot -2\right), x\right), s\right), x\right) \]

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f32 36.6%

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

   8. Simplified 36.6%

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

   9. Taylor expanded in s around inf

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

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

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

       2. unpow2 N/A

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

       3. associate-/l* N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f32 71.6%

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

   11. Simplified 71.6%

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

   if -1e-15 < 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 48.5%

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

Alternative 9: 48.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999943633011 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 4.999999943633011e-27) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999943633011e-27f) {
		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 <= 4.999999943633011e-27) 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(4.999999943633011e-27))
		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(4.999999943633011e-27))
		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 < 4.99999994e-27

   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 58.7%

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

   5. Simplified 58.7%

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

   if 4.99999994e-27 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 35.1%

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

Alternative 10: 46.8% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000036274937e-15) (/ -1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000036274937e-15f) {
		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 <= (-1.0000000036274937e-15)) 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(-1.0000000036274937e-15))
		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(-1.0000000036274937e-15))
		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 < -1e-15

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 42.2%

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

   5. Simplified 42.2%

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

   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 37.2%

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

   8. Simplified 37.2%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   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 41.8%

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

   10. Applied egg-rr 41.8%

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

   if -1e-15 < 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 48.5%

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

Alternative 11: 45.4% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000036274937e-15) (- 0.0 (/ s x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000036274937e-15f) {
		tmp = 0.0f - (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 <= (-1.0000000036274937e-15)) then
        tmp = 0.0e0 - (s / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000036274937e-15))
		tmp = Float32(Float32(0.0) - Float32(s / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000036274937e-15))
		tmp = single(0.0) - (s / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-15

   1. Initial program 100.0%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   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 42.2%

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

   5. Simplified 42.2%

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

   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 37.2%

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

   8. Simplified 37.2%

      \[\leadsto \color{blue}{0 - \frac{s}{x}} \]
   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 37.2%

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

   10. Applied egg-rr 37.2%

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

   if -1e-15 < 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 48.5%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;0 - \frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 99.9%

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

3. Taylor expanded in x around 0

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

5. Simplified 36.0%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))