Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.2s

Alternatives: 10

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 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 81.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4000:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \mathbf{elif}\;x \leq -2.000000047484456 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -4000.0)
   (/ 1.0 (* x (+ (/ 2.0 x) (/ -1.0 s))))
   (if (<= x -2.000000047484456e-33)
     (/ 1.0 (/ (- 4.0 (* x (/ (/ x s) s))) (+ 2.0 (/ x s))))
     (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s))))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -4000.0f) {
		tmp = 1.0f / (x * ((2.0f / x) + (-1.0f / s)));
	} else if (x <= -2.000000047484456e-33f) {
		tmp = 1.0f / ((4.0f - (x * ((x / s) / s))) / (2.0f + (x / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-4000.0e0)) then
        tmp = 1.0e0 / (x * ((2.0e0 / x) + ((-1.0e0) / s)))
    else if (x <= (-2.000000047484456e-33)) then
        tmp = 1.0e0 / ((4.0e0 - (x * ((x / s) / s))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-4000.0))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(2.0) / x) + Float32(Float32(-1.0) / s))));
	elseif (x <= Float32(-2.000000047484456e-33))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x * Float32(Float32(x / s) / s))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-4000.0))
		tmp = single(1.0) / (x * ((single(2.0) / x) + (single(-1.0) / s)));
	elseif (x <= single(-2.000000047484456e-33))
		tmp = single(1.0) / ((single(4.0) - (x * ((x / s) / s))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -4e3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 60.5%

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

      1. mul-1-neg 60.5%

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

      2. unsub-neg 60.5%

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

   5. Simplified 60.5%

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

   6. Taylor expanded in x around inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. metadata-eval 60.5%

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

   8. Simplified 60.5%

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

   if -4e3 < x < -2.00000005e-33

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 47.0%

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

      1. mul-1-neg 47.0%

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

      2. unsub-neg 47.0%

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

   5. Simplified 47.0%

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

      1. sub-neg 47.0%

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

      2. flip-+ 58.3%

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

      3. metadata-eval 58.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{4} - \left(-\frac{x}{s}\right) \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      4. distribute-neg-frac2 58.3%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{-s}} \cdot \left(-\frac{x}{s}\right)}{2 - \left(-\frac{x}{s}\right)}} \]

      5. distribute-neg-frac2 58.3%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \color{blue}{\frac{x}{-s}}}{2 - \left(-\frac{x}{s}\right)}} \]

      6. distribute-neg-frac2 58.3%

         \[\leadsto \frac{1}{\frac{4 - \frac{x}{-s} \cdot \frac{x}{-s}}{2 - \color{blue}{\frac{x}{-s}}}} \]

   7. Applied egg-rr 58.3%

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

      1. associate-*r/ 58.3%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{-s} \cdot x}{-s}}}{2 - \frac{x}{-s}}} \]

      2. add-sqr-sqrt -0.0%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}}{2 - \frac{x}{-s}}} \]

      3. sqrt-unprod 64.2%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}}{2 - \frac{x}{-s}}} \]

      4. sqr-neg 64.2%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\sqrt{\color{blue}{s \cdot s}}}}{2 - \frac{x}{-s}}} \]

      5. sqrt-unprod 56.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}}{2 - \frac{x}{-s}}} \]

      6. add-sqr-sqrt 56.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\color{blue}{s}}}{2 - \frac{x}{-s}}} \]

      7. remove-double-neg 56.6%

         \[\leadsto \frac{1}{\frac{4 - \frac{\frac{x}{-s} \cdot x}{\color{blue}{-\left(-s\right)}}}{2 - \frac{x}{-s}}} \]

      8. distribute-neg-frac2 56.6%

         \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{\frac{x}{-s} \cdot x}{-s}\right)}}{2 - \frac{x}{-s}}} \]

      9. associate-*r/ 56.6%

         \[\leadsto \frac{1}{\frac{4 - \left(-\color{blue}{\frac{x}{-s} \cdot \frac{x}{-s}}\right)}{2 - \frac{x}{-s}}} \]

      10. distribute-lft-neg-in 56.6%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\left(-\frac{x}{-s}\right) \cdot \frac{x}{-s}}}{2 - \frac{x}{-s}}} \]

      11. distribute-frac-neg2 56.6%

          \[\leadsto \frac{1}{\frac{4 - \left(-\color{blue}{\left(-\frac{x}{s}\right)}\right) \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}} \]

      12. remove-double-neg 56.6%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{s}} \cdot \frac{x}{-s}}{2 - \frac{x}{-s}}} \]

      13. distribute-frac-neg2 56.6%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \color{blue}{\left(-\frac{x}{s}\right)}}{2 - \frac{x}{-s}}} \]

      14. distribute-frac-neg 56.6%

          \[\leadsto \frac{1}{\frac{4 - \frac{x}{s} \cdot \color{blue}{\frac{-x}{s}}}{2 - \frac{x}{-s}}} \]

      15. associate-*r/ 56.6%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{\frac{x}{s} \cdot \left(-x\right)}{s}}}{2 - \frac{x}{-s}}} \]

   9. Applied egg-rr 56.6%

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

       1. *-commutative 56.6%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\left(-x\right) \cdot \frac{x}{s}}}{s}}{2 - \frac{x}{-s}}} \]

       2. *-un-lft-identity 56.6%

          \[\leadsto \frac{1}{\frac{4 - \frac{\left(-x\right) \cdot \frac{x}{s}}{\color{blue}{1 \cdot s}}}{2 - \frac{x}{-s}}} \]

       3. times-frac 64.0%

          \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{-x}{1} \cdot \frac{\frac{x}{s}}{s}}}{2 - \frac{x}{-s}}} \]

       4. add-sqr-sqrt 64.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       5. sqrt-unprod 61.4%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       6. sqr-neg 61.4%

          \[\leadsto \frac{1}{\frac{4 - \frac{\sqrt{\color{blue}{x \cdot x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       7. sqrt-unprod -0.0%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

       8. add-sqr-sqrt 65.7%

          \[\leadsto \frac{1}{\frac{4 - \frac{\color{blue}{x}}{1} \cdot \frac{\frac{x}{s}}{s}}{2 - \frac{x}{-s}}} \]

   11. Applied egg-rr 65.7%

       \[\leadsto \frac{1}{\frac{4 - \color{blue}{\frac{x}{1} \cdot \frac{\frac{x}{s}}{s}}}{2 - \frac{x}{-s}}} \]

   if -2.00000005e-33 < x

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 93.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4000:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \mathbf{elif}\;x \leq -2.000000047484456 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{\frac{4 - x \cdot \frac{\frac{x}{s}}{s}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 48.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.0000000031710769e-28)
   0.5
   (/ 1.0 (* x (+ (/ 2.0 x) (/ -1.0 s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.0000000031710769e-28f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (x * ((2.0f / x) + (-1.0f / 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 <= (-1.0000000031710769e-28)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (x * ((2.0e0 / x) + ((-1.0e0) / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.0000000031710769e-28))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(2.0) / x) + Float32(Float32(-1.0) / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.0000000031710769e-28))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (x * ((single(2.0) / x) + (single(-1.0) / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < -1e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 34.9%

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

   if -1e-28 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   5. Simplified 60.1%

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

   6. Taylor expanded in x around inf 60.5%

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

      1. associate-*r/ 60.5%

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

      2. metadata-eval 60.5%

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

   8. Simplified 60.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.0000000031710769e-28)
   (/ 1.0 (* x (+ (/ 2.0 x) (/ -1.0 s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0000000031710769e-28f) {
		tmp = 1.0f / (x * ((2.0f / x) + (-1.0f / s)));
	} else {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.0000000031710769e-28)) then
        tmp = 1.0e0 / (x * ((2.0e0 / x) + ((-1.0e0) / s)))
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.0000000031710769e-28))
		tmp = Float32(Float32(1.0) / Float32(x * Float32(Float32(Float32(2.0) / x) + Float32(Float32(-1.0) / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.0000000031710769e-28))
		tmp = single(1.0) / (x * ((single(2.0) / x) + (single(-1.0) / s)));
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1e-28

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 51.8%

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

      1. mul-1-neg 51.8%

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

      2. unsub-neg 51.8%

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

   5. Simplified 51.8%

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

   6. Taylor expanded in x around inf 52.6%

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

      1. associate-*r/ 52.6%

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

      2. metadata-eval 52.6%

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

   8. Simplified 52.6%

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

   if -1e-28 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 92.2%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{x \cdot \left(\frac{2}{x} + \frac{-1}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.0000000031710769e-28) 0.5 (/ 1.0 (- 2.0 (/ x s)))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= -1.0000000031710769e-28f) {
		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 <= (-1.0000000031710769e-28)) 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(-x) <= Float32(-1.0000000031710769e-28))
		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 <= single(-1.0000000031710769e-28))
		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 (neg.f32 x) < -1e-28

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 34.9%

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

   if -1e-28 < (neg.f32 x)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 60.1%

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

      1. mul-1-neg 60.1%

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

      2. unsub-neg 60.1%

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

   5. Simplified 60.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq -1.0000000031710769 \cdot 10^{-28}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.1% accurate, 9.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.1%

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

   if 1 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.3%

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

      1. mul-1-neg 55.3%

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

      2. unsub-neg 55.3%

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

   5. Simplified 55.3%

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

   6. Taylor expanded in x around inf 47.7%

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

      1. associate-*r/ 47.7%

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

      2. neg-mul-1 47.7%

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

   8. Simplified 47.7%

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

      1. clear-num 55.3%

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

      2. inv-pow 55.3%

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

      3. add-sqr-sqrt -0.0%

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

      4. sqrt-unprod 62.9%

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

      5. sqr-neg 62.9%

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

      6. sqrt-unprod 55.3%

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

      7. add-sqr-sqrt 55.3%

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

   10. Applied egg-rr 55.3%

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

       1. unpow-1 55.3%

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

   12. Simplified 55.3%

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

4. Final simplification 47.7%

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

Alternative 7: 48.1% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 3.9999998989515007e-5) 0.5 (/ -1.0 (/ x s))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 3.9999998989515007e-5f) {
		tmp = 0.5f;
	} else {
		tmp = -1.0f / (x / s);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 3.9999998989515007e-5) then
        tmp = 0.5e0
    else
        tmp = (-1.0e0) / (x / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(3.9999998989515007e-5))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(-1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(3.9999998989515007e-5))
		tmp = single(0.5);
	else
		tmp = single(-1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 3.9999999e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.2%

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

   if 3.9999999e-5 < (neg.f32 x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. unsub-neg 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 51.7%

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

      1. mul-1-neg 51.7%

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

      2. distribute-frac-neg2 51.7%

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

   8. Simplified 51.7%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 3.9999998989515007 \cdot 10^{-5}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.00019999999494757503) (/ s (- x)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.00019999999494757503f) {
		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 <= (-0.00019999999494757503e0)) 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(-0.00019999999494757503))
		tmp = Float32(s / Float32(-x));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.00019999999494757503))
		tmp = s / -x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999995e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 51.7%

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

      1. mul-1-neg 51.7%

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

      2. unsub-neg 51.7%

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

   5. Simplified 51.7%

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

   6. Taylor expanded in x around inf 44.7%

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

      1. associate-*r/ 44.7%

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

      2. neg-mul-1 44.7%

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

   8. Simplified 44.7%

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

   if -1.99999995e-4 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 46.2%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00019999999494757503:\\ \;\;\;\;\frac{s}{-x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.7% accurate, 13.4× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.0f) {
		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 <= (-1.0e0)) 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(-1.0))
		tmp = 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(-1.0))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 55.3%

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

      1. mul-1-neg 55.3%

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

      2. unsub-neg 55.3%

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

   5. Simplified 55.3%

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

   6. Taylor expanded in x around inf 47.7%

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

      1. associate-*r/ 47.7%

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

      2. neg-mul-1 47.7%

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

   8. Simplified 47.7%

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

      1. div-inv 47.7%

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

      2. add-sqr-sqrt -0.0%

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

      3. sqrt-unprod 61.1%

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

      4. sqr-neg 61.1%

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

      5. sqrt-unprod 47.7%

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

      6. add-sqr-sqrt 47.7%

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

   10. Applied egg-rr 47.7%

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

       1. associate-*r/ 47.7%

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

       2. *-rgt-identity 47.7%

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

   12. Simplified 47.7%

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

   if -1 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 45.1%

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

4. Final simplification 45.8%

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

Alternative 10: 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 99.8%

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

3. Taylor expanded in x around 0 35.3%

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

4. Final simplification 35.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))