Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 64.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.0020000000949949026:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{s} \cdot \frac{1}{s}\right)\right)\right) + \left(2 - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.0020000000949949026)
   0.5
   (/ 1.0 (+ (* 0.5 (* x (* x (* (/ 1.0 s) (/ 1.0 s))))) (- 2.0 (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.0020000000949949026f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((0.5f * (x * (x * ((1.0f / s) * (1.0f / s))))) + (2.0f - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.0020000000949949026e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((0.5e0 * (x * (x * ((1.0e0 / s) * (1.0e0 / s))))) + (2.0e0 - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.0020000000949949026))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(Float32(1.0) / s) * Float32(Float32(1.0) / s))))) + Float32(Float32(2.0) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.0020000000949949026))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(0.5) * (x * (x * ((single(1.0) / s) * (single(1.0) / s))))) + (single(2.0) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.00200000009

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.7%

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

   if 0.00200000009 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. associate-+l+ 74.9%

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

      3. unpow2 74.9%

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

      4. unpow2 74.9%

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

      5. times-frac 67.6%

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

      6. +-commutative 67.6%

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

      7. mul-1-neg 67.6%

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

      8. unsub-neg 67.6%

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

   4. Simplified 67.6%

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

      1. clear-num 63.3%

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

      2. frac-times 66.7%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\frac{1 \cdot x}{\frac{s}{x} \cdot s}} + \frac{-x}{s}} \]

      3. *-un-lft-identity 66.7%

         \[\leadsto \frac{1}{0.5 \cdot \frac{\color{blue}{x}}{\frac{s}{x} \cdot s} + \frac{-x}{s}} \]

   6. Applied egg-rr 71.0%

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

      1. *-un-lft-identity 66.7%

         \[\leadsto \frac{1}{0.5 \cdot \frac{\color{blue}{1 \cdot x}}{\frac{s}{x} \cdot s} + \frac{-x}{s}} \]

      2. frac-times 63.3%

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

      3. clear-num 63.3%

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

      4. div-inv 63.3%

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

      5. div-inv 63.3%

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

      6. swap-sqr 72.3%

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

   8. Applied egg-rr 75.8%

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

      1. associate-*l* 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.0020000000949949026:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \left(x \cdot \left(\frac{1}{s} \cdot \frac{1}{s}\right)\right)\right) + \left(2 - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 3: 64.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \left(x \cdot \frac{\frac{1}{s}}{s}\right)\right) - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   0.5
   (/ 1.0 (- (* 0.5 (* x (* x (/ (/ 1.0 s) s)))) (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / ((0.5f * (x * (x * ((1.0f / s) / s)))) - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.10000000149011612e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / ((0.5e0 * (x * (x * ((1.0e0 / s) / s)))) - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(Float32(1.0) / s) / s)))) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		tmp = single(0.5);
	else
		tmp = single(1.0) / ((single(0.5) * (x * (x * ((single(1.0) / s) / s)))) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 0.100000001

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.5%

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

   if 0.100000001 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-+l+ 75.0%

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

      3. unpow2 75.0%

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

      4. unpow2 75.0%

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

      5. times-frac 66.2%

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

      6. +-commutative 66.2%

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

      7. mul-1-neg 66.2%

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

      8. unsub-neg 66.2%

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

   4. Simplified 66.2%

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

   5. Taylor expanded in x around inf 66.1%

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

      1. neg-mul-1 32.8%

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

      2. distribute-neg-frac 32.8%

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

   7. Simplified 66.1%

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

      1. clear-num 66.1%

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

      2. frac-times 69.7%

         \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\frac{1 \cdot x}{\frac{s}{x} \cdot s}} + \frac{-x}{s}} \]

      3. *-un-lft-identity 69.7%

         \[\leadsto \frac{1}{0.5 \cdot \frac{\color{blue}{x}}{\frac{s}{x} \cdot s} + \frac{-x}{s}} \]

   9. Applied egg-rr 69.7%

      \[\leadsto \frac{1}{0.5 \cdot \color{blue}{\frac{x}{\frac{s}{x} \cdot s}} + \frac{-x}{s}} \]
   10. Step-by-step derivation

       1. *-un-lft-identity 69.7%

          \[\leadsto \frac{1}{0.5 \cdot \frac{\color{blue}{1 \cdot x}}{\frac{s}{x} \cdot s} + \frac{-x}{s}} \]

       2. frac-times 66.1%

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

       3. clear-num 66.1%

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

       4. div-inv 66.1%

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

       5. div-inv 66.1%

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

       6. swap-sqr 75.9%

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

   11. Applied egg-rr 75.9%

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

       1. associate-*l* 81.2%

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

       2. associate-*r/ 81.2%

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

       3. *-rgt-identity 81.2%

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

   13. Simplified 81.2%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5 \cdot \left(x \cdot \left(x \cdot \frac{\frac{1}{s}}{s}\right)\right) - \frac{x}{s}}\\ \end{array} \]

Alternative 4: 63.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. associate-+l+ 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. times-frac 66.5%

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

      6. +-commutative 66.5%

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

      7. mul-1-neg 66.5%

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

      8. unsub-neg 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. unpow2 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 64.8%

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

   7. Simplified 64.8%

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

      1. associate-*r* 64.8%

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

      2. metadata-eval 64.8%

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

      3. times-frac 64.8%

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

      4. *-commutative 64.8%

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

      5. *-un-lft-identity 64.8%

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

      6. times-frac 74.3%

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

      7. *-commutative 74.3%

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

      8. clear-num 75.4%

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

      9. associate-*r* 75.4%

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

      10. times-frac 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 62.1%

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

Alternative 5: 58.9% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. associate-+l+ 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. times-frac 66.5%

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

      6. +-commutative 66.5%

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

      7. mul-1-neg 66.5%

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

      8. unsub-neg 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. unpow2 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 56.3%

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

Alternative 6: 60.8% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{\frac{2}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 40000000.0) 0.5 (* (* s s) (/ (/ 2.0 x) x))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 40000000.0f) {
		tmp = 0.5f;
	} else {
		tmp = (s * s) * ((2.0f / x) / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 40000000.0e0) then
        tmp = 0.5e0
    else
        tmp = (s * s) * ((2.0e0 / x) / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(40000000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(s * s) * Float32(Float32(Float32(2.0) / x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(40000000.0))
		tmp = single(0.5);
	else
		tmp = (s * s) * ((single(2.0) / x) / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 4e7

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 48.4%

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

   if 4e7 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 84.6%

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

      1. +-commutative 84.6%

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

      2. associate-+l+ 84.6%

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

      3. unpow2 84.6%

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

      4. unpow2 84.6%

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

      5. times-frac 75.1%

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

      6. +-commutative 75.1%

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

      7. mul-1-neg 75.1%

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

      8. unsub-neg 75.1%

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

   4. Simplified 75.1%

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

   5. Taylor expanded in x around inf 83.3%

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

      1. associate-*r/ 83.3%

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

      2. unpow2 83.3%

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

      3. unpow2 83.3%

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

   7. Simplified 83.3%

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

   8. Taylor expanded in s around 0 83.3%

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

      1. unpow2 83.3%

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

      2. associate-*r/ 83.3%

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

      3. *-commutative 83.3%

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

      4. associate-*r/ 83.3%

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

      5. unpow2 83.3%

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

      6. associate-/r* 83.3%

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

   10. Simplified 83.3%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 40000000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(s \cdot s\right) \cdot \frac{\frac{2}{x}}{x}\\ \end{array} \]

Alternative 7: 63.4% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. associate-+l+ 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. times-frac 66.5%

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

      6. +-commutative 66.5%

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

      7. mul-1-neg 66.5%

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

      8. unsub-neg 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. associate-*r/ 74.3%

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

      2. unpow2 74.3%

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

      3. unpow2 74.3%

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

   7. Simplified 74.3%

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

      1. *-commutative 74.3%

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

      2. times-frac 79.7%

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

   9. Applied egg-rr 79.7%

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

4. Final simplification 61.7%

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

Alternative 8: 63.4% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.4%

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

      1. +-commutative 75.4%

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

      2. associate-+l+ 75.4%

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

      3. unpow2 75.4%

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

      4. unpow2 75.4%

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

      5. times-frac 66.5%

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

      6. +-commutative 66.5%

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

      7. mul-1-neg 66.5%

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

      8. unsub-neg 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. unpow2 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 64.8%

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

   7. Simplified 64.8%

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

      1. associate-*r* 64.8%

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

      2. metadata-eval 64.8%

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

      3. times-frac 64.8%

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

      4. *-commutative 64.8%

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

      5. *-un-lft-identity 64.8%

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

      6. times-frac 74.3%

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

      7. *-commutative 74.3%

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

      8. associate-/r* 79.7%

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

      9. associate-*r* 79.7%

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

   9. Applied egg-rr 79.7%

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

4. Final simplification 61.7%

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

Alternative 9: 49.9% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) -5.0) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -5.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= (-5.0e0)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(-5.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 28.1%

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

   if -5 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 57.7%

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

      1. mul-1-neg 57.7%

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

      2. unsub-neg 57.7%

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

   4. Simplified 57.7%

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

4. Final simplification 46.2%

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

Alternative 10: 48.5% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s))) (if (<= t_0 2.0) 0.5 (/ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / t_0;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(2.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(2.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 51.4%

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

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

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 32.9%

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

      1. mul-1-neg 32.9%

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

      2. unsub-neg 32.9%

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

   4. Simplified 32.9%

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

   5. Taylor expanded in x around inf 32.9%

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

      1. neg-mul-1 32.9%

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

      2. distribute-neg-frac 32.9%

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

   7. Simplified 32.9%

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

4. Final simplification 44.7%

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

Alternative 11: 46.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000233721948e-7) (* s (/ -1.0 x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000233721948e-7f) {
		tmp = s * (-1.0f / 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 <= (-2.0000000233721948e-7)) then
        tmp = s * ((-1.0e0) / x)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0000000233721948e-7))
		tmp = Float32(s * Float32(Float32(-1.0) / x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000233721948e-7))
		tmp = s * (single(-1.0) / x);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. unsub-neg 40.3%

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

   4. Simplified 40.3%

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

   5. Taylor expanded in x around inf 36.7%

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

      1. mul-1-neg 36.7%

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

   7. Simplified 36.7%

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

      1. add-sqr-sqrt -0.0%

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

      2. sqrt-prod 45.2%

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

      3. sqr-neg 45.2%

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

      4. sqrt-unprod 36.5%

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

      5. add-sqr-sqrt 36.5%

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

      6. clear-num 39.5%

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

      7. associate-/r/ 36.5%

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

      8. add-sqr-sqrt 36.5%

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

      9. sqrt-unprod 45.2%

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

      10. sqr-neg 45.2%

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

      11. sqrt-prod -0.0%

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

      12. add-sqr-sqrt 36.7%

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

   9. Applied egg-rr 36.7%

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

   if -2.00000002e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.3%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;s \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 12: 46.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.0000000233721948e-7) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.0000000233721948e-7f) {
		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 <= (-2.0000000233721948e-7)) 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(-2.0000000233721948e-7))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-2.0000000233721948e-7))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000002e-7

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. unsub-neg 40.3%

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

   4. Simplified 40.3%

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

   5. Taylor expanded in x around inf 36.7%

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

      1. mul-1-neg 36.7%

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

   7. Simplified 36.7%

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

   if -2.00000002e-7 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 46.3%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.0000000233721948 \cdot 10^{-7}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 13: 35.7% 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. Taylor expanded in x around 0 35.1%

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

3. Final simplification 35.1%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))