Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 7.5s

Alternatives: 9

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: 63.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot -0.5 + \frac{s}{\frac{x}{x}}}{s \cdot \left(-s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999796611898e-32)
   (/ 1.0 (+ 2.0 (* x (/ (+ (* x -0.5) (/ s (/ x x))) (* s (- s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = 1.0f / (2.0f + (x * (((x * -0.5f) + (s / (x / x))) / (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 <= (-9.999999796611898e-32)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((x * (-0.5e0)) + (s / (x / x))) / (s * -s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(x * Float32(-0.5)) + Float32(s / Float32(x / x))) / Float32(s * Float32(-s))))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999796611898e-32))
		tmp = single(1.0) / (single(2.0) + (x * (((x * single(-0.5)) + (s / (x / x))) / (s * -s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

      3. unpow2 73.8%

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

      4. unpow2 73.8%

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

      5. times-frac 68.2%

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

   4. Simplified 68.2%

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

      1. associate-*l/ 68.2%

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

      2. associate-/l* 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*r/ 71.7%

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

      2. associate-/r/ 71.7%

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

      3. times-frac 67.7%

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

      4. clear-num 67.7%

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

      5. div-inv 67.7%

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

      6. frac-2neg 67.7%

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

      7. frac-sub 71.7%

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

      8. div-inv 71.7%

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

      9. clear-num 71.7%

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

      10. distribute-rgt-neg-in 71.7%

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

      11. associate-/r/ 71.7%

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

      12. div-inv 71.7%

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

      13. clear-num 71.7%

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

      14. distribute-rgt-neg-in 71.7%

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

      15. distribute-neg-frac 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. associate-*r/ 77.1%

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

      2. associate-/r/ 78.3%

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

      3. sub-neg 78.3%

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

      4. distribute-rgt-neg-out 78.3%

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

      5. *-commutative 78.3%

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

      6. distribute-rgt-neg-in 78.3%

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

      7. *-commutative 78.3%

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

      8. distribute-rgt-neg-in 78.3%

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

      9. metadata-eval 78.3%

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

      10. distribute-rgt-neg-out 78.3%

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

      11. remove-double-neg 78.3%

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

      12. associate-*l/ 78.3%

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

      13. associate-/l* 78.3%

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

   10. Simplified 78.3%

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

   11. Taylor expanded in s around 0 78.3%

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

       1. *-commutative 78.3%

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

   13. Simplified 78.3%

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

   if -9.9999998e-32 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 44.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + x \cdot \frac{x \cdot -0.5 + \frac{s}{\frac{x}{x}}}{s \cdot \left(-s\right)}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 3: 63.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{x \cdot 0.5}{s \cdot s} - \frac{1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999796611898e-32)
   (/ 1.0 (+ 2.0 (* x (- (/ (* x 0.5) (* s s)) (/ 1.0 s)))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = 1.0f / (2.0f + (x * (((x * 0.5f) / (s * s)) - (1.0f / s))));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-9.999999796611898e-32)) then
        tmp = 1.0e0 / (2.0e0 + (x * (((x * 0.5e0) / (s * s)) - (1.0e0 / s))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(x * Float32(Float32(Float32(x * Float32(0.5)) / Float32(s * s)) - Float32(Float32(1.0) / s)))));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-9.999999796611898e-32))
		tmp = single(1.0) / (single(2.0) + (x * (((x * single(0.5)) / (s * s)) - (single(1.0) / s))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

      3. unpow2 73.8%

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

      4. unpow2 73.8%

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

      5. times-frac 68.2%

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

   4. Simplified 68.2%

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

      1. associate-*l/ 68.2%

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

      2. associate-/l* 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. associate-*r/ 71.7%

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

      2. associate-/r/ 71.7%

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

      3. times-frac 67.7%

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

      4. clear-num 67.7%

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

      5. div-inv 67.7%

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

      6. frac-2neg 67.7%

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

      7. frac-sub 71.7%

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

      8. div-inv 71.7%

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

      9. clear-num 71.7%

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

      10. distribute-rgt-neg-in 71.7%

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

      11. associate-/r/ 71.7%

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

      12. div-inv 71.7%

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

      13. clear-num 71.7%

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

      14. distribute-rgt-neg-in 71.7%

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

      15. distribute-neg-frac 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. associate-*r/ 77.1%

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

      2. associate-/r/ 78.3%

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

      3. sub-neg 78.3%

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

      4. distribute-rgt-neg-out 78.3%

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

      5. *-commutative 78.3%

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

      6. distribute-rgt-neg-in 78.3%

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

      7. *-commutative 78.3%

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

      8. distribute-rgt-neg-in 78.3%

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

      9. metadata-eval 78.3%

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

      10. distribute-rgt-neg-out 78.3%

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

      11. remove-double-neg 78.3%

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

      12. associate-*l/ 78.3%

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

      13. associate-/l* 78.3%

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

   10. Simplified 78.3%

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

   11. Taylor expanded in s around 0 78.3%

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

       1. associate-*r/ 78.3%

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

       2. *-commutative 78.3%

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

       3. unpow2 78.3%

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

   13. Simplified 78.3%

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

   if -9.9999998e-32 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 44.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + x \cdot \left(\frac{x \cdot 0.5}{s \cdot s} - \frac{1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4: 62.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -9.999999796611898e-32)
   (/ 1.0 (+ 2.0 (* 0.5 (* x (/ x (* s s))))))
   0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -9.999999796611898e-32f) {
		tmp = 1.0f / (2.0f + (0.5f * (x * (x / (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 <= (-9.999999796611898e-32)) then
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * (x * (x / (s * s)))))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-9.999999796611898e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(x * Float32(x / 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(-9.999999796611898e-32))
		tmp = single(1.0) / (single(2.0) + (single(0.5) * (x * (x / (s * s)))));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -9.9999998e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 73.8%

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

      1. mul-1-neg 73.8%

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

      2. unsub-neg 73.8%

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

      3. unpow2 73.8%

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

      4. unpow2 73.8%

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

      5. times-frac 68.2%

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

   4. Simplified 68.2%

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

      1. associate-*l/ 68.2%

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

      2. associate-/l* 71.7%

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

   6. Applied egg-rr 71.7%

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

   7. Taylor expanded in x around inf 72.9%

      \[\leadsto \frac{1}{2 + \color{blue}{0.5 \cdot \frac{{x}^{2}}{{s}^{2}}}} \]
   8. Step-by-step derivation

      1. unpow2 72.9%

         \[\leadsto \frac{1}{2 + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{s}^{2}}} \]

      2. unpow2 72.9%

         \[\leadsto \frac{1}{2 + 0.5 \cdot \frac{x \cdot x}{\color{blue}{s \cdot s}}} \]

      3. associate-*r/ 77.4%

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

   9. Simplified 77.4%

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

   if -9.9999998e-32 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 44.7%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.999999796611898 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5: 56.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -5.0000000843119176e-17) (* 2.0 (* (/ s x) (/ s x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -5.0000000843119176e-17f) {
		tmp = 2.0f * ((s / x) * (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 <= (-5.0000000843119176e-17)) then
        tmp = 2.0e0 * ((s / x) * (s / x))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-5.0000000843119176e-17))
		tmp = Float32(Float32(2.0) * Float32(Float32(s / x) * 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(-5.0000000843119176e-17))
		tmp = single(2.0) * ((s / x) * (s / x));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -5.00000008e-17

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 75.3%

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

      1. mul-1-neg 75.3%

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

      2. unsub-neg 75.3%

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

      3. unpow2 75.3%

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

      4. unpow2 75.3%

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

      5. times-frac 71.2%

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

   4. Simplified 71.2%

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

   5. Taylor expanded in x around inf 70.6%

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

      1. unpow2 70.6%

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

      2. unpow2 70.6%

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

      3. times-frac 64.1%

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

   7. Simplified 64.1%

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

   if -5.00000008e-17 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 44.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \left(\frac{s}{x} \cdot \frac{s}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 6: 58.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.000000031374395e-22) (* 2.0 (/ (* s s) (* x x))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.000000031374395e-22f) {
		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 <= (-1.000000031374395e-22)) 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(-1.000000031374395e-22))
		tmp = Float32(Float32(2.0) * Float32(Float32(s * 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(-1.000000031374395e-22))
		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 < -1.00000003e-22

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. unsub-neg 75.9%

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

      3. unpow2 75.9%

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

      4. unpow2 75.9%

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

      5. times-frac 68.9%

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

   4. Simplified 68.9%

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

      1. associate-*l/ 68.9%

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

      2. associate-/l* 71.9%

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

   6. Applied egg-rr 71.9%

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

   7. Taylor expanded in x around inf 69.7%

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

      1. unpow2 69.7%

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

      2. unpow2 69.7%

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

   9. Simplified 69.7%

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

   if -1.00000003e-22 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 45.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{s \cdot s}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7: 49.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.000000023742228e-32) (/ 1.0 (- 2.0 (/ x s))) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.000000023742228e-32f) {
		tmp = 1.0f / (2.0f - (x / s));
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 1.000000023742228e-32) then
        tmp = 1.0e0 / (2.0e0 - (x / s))
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.000000023742228e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.000000023742228e-32))
		tmp = single(1.0) / (single(2.0) - (x / s));
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000002e-32

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 45.0%

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

      1. mul-1-neg 45.0%

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

      2. unsub-neg 45.0%

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

   4. Simplified 45.0%

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

   if 1.00000002e-32 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 38.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.000000023742228 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8: 46.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.000000026702864e-10) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.000000026702864e-10f) {
		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 <= (-2.000000026702864e-10)) 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(-2.000000026702864e-10))
		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(-2.000000026702864e-10))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2.00000003e-10

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 38.5%

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

      1. mul-1-neg 38.5%

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

      2. unsub-neg 38.5%

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

   4. Simplified 38.5%

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

   5. Taylor expanded in x around inf 37.6%

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

      1. associate-*r/ 37.6%

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

      2. neg-mul-1 37.6%

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

   7. Simplified 37.6%

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

   if -2.00000003e-10 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 42.6%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9: 35.6% 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 29.6%

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

3. Final simplification 29.6%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))