Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

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. Add Preprocessing

3. Final simplification 99.9%

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

Alternative 2: 62.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 5.999999941330714e-10)
     0.5
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ x (/ (* s (- s)) x))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 5.999999941330714e-10f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x / ((s * -s) / x))) / (2.0f + (x / s)));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 5.999999941330714e-10) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / ((s * -s) / x))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (x / s)
    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(5.999999941330714e-10))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(Float32(s * Float32(-s)) / x))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(5.999999941330714e-10))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x / ((s * -s) / x))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < 5.99999994e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 49.4%

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

   if 5.99999994e-10 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 22.0%

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

      1. mul-1-neg 22.0%

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

      2. unsub-neg 22.0%

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

   5. Simplified 22.0%

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

      1. sub-neg 22.0%

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

      2. flip-+ 59.8%

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

      3. metadata-eval 59.8%

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

      4. distribute-neg-frac 59.8%

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

      5. distribute-neg-frac 59.8%

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

      6. distribute-neg-frac 59.8%

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

   7. Applied egg-rr 59.8%

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

      1. clear-num 48.0%

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

      2. frac-2neg 48.0%

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

      3. frac-times 51.5%

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

      4. *-un-lft-identity 51.5%

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

      5. remove-double-neg 51.5%

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

      6. add-sqr-sqrt 51.5%

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

      7. sqrt-unprod 51.4%

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

      8. sqr-neg 51.4%

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

      9. sqrt-unprod -0.0%

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

      10. add-sqr-sqrt 51.5%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 63.3%

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

      13. sqr-neg 63.3%

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

      14. sqrt-unprod 51.5%

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

      15. add-sqr-sqrt 51.5%

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

   9. Applied egg-rr 63.2%

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

       1. frac-2neg 63.2%

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

       2. associate-*l/ 73.7%

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

       3. add-sqr-sqrt 73.7%

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

       4. sqrt-unprod 66.3%

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

       5. sqr-neg 66.3%

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

       6. sqrt-unprod -0.0%

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

       7. add-sqr-sqrt 73.6%

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

   11. Applied egg-rr 73.6%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   8. Simplified 100.0%

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

      1. associate-/r/ 87.9%

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

      2. add-sqr-sqrt 87.9%

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

      3. sqrt-unprod 92.2%

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

      4. sqr-neg 92.2%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 87.9%

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

   10. Applied egg-rr 87.9%

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

       1. associate-/r/ 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 5.999999941330714 \cdot 10^{-10}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{\frac{s \cdot \left(-s\right)}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -1.0)
     0.5
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (+ 2.0 (/ x s))))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (2.0f + (x / s)));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= (-1.0e0)) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (2.0e0 + (x / s)))
    else
        tmp = 1.0e0 / (x / s)
    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(-1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(Float32(2.0) + Float32(x / s))));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = -x / s;
	tmp = single(0.0);
	if (t_0 <= single(-1.0))
		tmp = single(0.5);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (single(2.0) + (x / s)));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if (/.f32 (neg.f32 x) s) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -1 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

   5. Simplified 50.7%

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

      1. sub-neg 50.7%

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

      2. flip-+ 74.4%

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

      3. metadata-eval 74.4%

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

      4. distribute-neg-frac 74.4%

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

      5. distribute-neg-frac 74.4%

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

      6. distribute-neg-frac 74.4%

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

   7. Applied egg-rr 74.4%

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

      1. clear-num 33.3%

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

      2. frac-2neg 33.3%

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

      3. frac-times 35.4%

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

      4. *-un-lft-identity 35.4%

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

      5. remove-double-neg 35.4%

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

      6. add-sqr-sqrt 33.0%

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

      7. sqrt-unprod 35.4%

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

      8. sqr-neg 35.4%

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

      9. sqrt-unprod 2.4%

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

      10. add-sqr-sqrt 35.4%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 42.6%

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

      13. sqr-neg 42.6%

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

      14. sqrt-unprod 35.4%

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

      15. add-sqr-sqrt 35.4%

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

   9. Applied egg-rr 76.6%

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

   10. Taylor expanded in x around 0 76.6%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   8. Simplified 100.0%

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

      1. associate-/r/ 87.9%

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

      2. add-sqr-sqrt 87.9%

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

      3. sqrt-unprod 92.2%

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

      4. sqr-neg 92.2%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 87.9%

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

   10. Applied egg-rr 87.9%

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

       1. associate-/r/ 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{2 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t_0 \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 2.0)
     0.5
     (if (<= t_0 1.9999999360571385e+38)
       (/ 1.0 (/ (- 4.0 (/ x (* s (/ s x)))) (/ x s)))
       (/ 1.0 (/ x s))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 2.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 1.9999999360571385e+38f) {
		tmp = 1.0f / ((4.0f - (x / (s * (s / x)))) / (x / s));
	} 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) :: t_0
    real(4) :: tmp
    t_0 = -x / s
    if (t_0 <= 2.0e0) then
        tmp = 0.5e0
    else if (t_0 <= 1.9999999360571385e+38) then
        tmp = 1.0e0 / ((4.0e0 - (x / (s * (s / x)))) / (x / s))
    else
        tmp = 1.0e0 / (x / s)
    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);
	elseif (t_0 <= Float32(1.9999999360571385e+38))
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(4.0) - Float32(x / Float32(s * Float32(s / x)))) / Float32(x / s)));
	else
		tmp = Float32(Float32(1.0) / Float32(x / s));
	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);
	elseif (t_0 <= single(1.9999999360571385e+38))
		tmp = single(1.0) / ((single(4.0) - (x / (s * (s / x)))) / (x / s));
	else
		tmp = single(1.0) / (x / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 50.8%

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

   if 2 < (/.f32 (neg.f32 x) s) < 1.99999994e38

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 11.4%

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

      1. mul-1-neg 11.4%

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

      2. unsub-neg 11.4%

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

   5. Simplified 11.4%

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

      1. sub-neg 11.4%

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

      2. flip-+ 56.0%

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

      3. metadata-eval 56.0%

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

      4. distribute-neg-frac 56.0%

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

      5. distribute-neg-frac 56.0%

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

      6. distribute-neg-frac 56.0%

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

   7. Applied egg-rr 56.0%

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

   8. Taylor expanded in x around inf 56.0%

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

      1. clear-num 56.0%

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

      2. frac-2neg 56.0%

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

      3. frac-times 60.1%

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

      4. *-un-lft-identity 60.1%

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

      5. remove-double-neg 60.1%

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

      6. add-sqr-sqrt 60.1%

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

      7. sqrt-unprod 59.9%

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

      8. sqr-neg 59.9%

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

      9. sqrt-unprod -0.0%

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

      10. add-sqr-sqrt 60.1%

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

      11. add-sqr-sqrt -0.0%

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

      12. sqrt-unprod 73.9%

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

      13. sqr-neg 73.9%

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

      14. sqrt-unprod 60.1%

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

      15. add-sqr-sqrt 60.1%

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

   10. Applied egg-rr 60.1%

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

   if 1.99999994e38 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-frac-neg 100.0%

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

   8. Simplified 100.0%

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

      1. associate-/r/ 87.9%

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

      2. add-sqr-sqrt 87.9%

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

      3. sqrt-unprod 92.2%

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

      4. sqr-neg 92.2%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 87.9%

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

   10. Applied egg-rr 87.9%

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

       1. associate-/r/ 100.0%

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

   12. Applied egg-rr 100.0%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 2:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 1.9999999360571385 \cdot 10^{+38}:\\ \;\;\;\;\frac{1}{\frac{4 - \frac{x}{s \cdot \frac{s}{x}}}{\frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -1.0f) {
		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) <= (-1.0e0)) 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(-1.0))
		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(-1.0))
		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) < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 28.1%

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

   if -1 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.1%

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

      1. mul-1-neg 59.1%

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

      2. unsub-neg 59.1%

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

   5. Simplified 59.1%

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

4. Final simplification 46.5%

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

Alternative 6: 48.4% accurate, 7.7× 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. Add Preprocessing

   3. Taylor expanded in x around 0 50.8%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. unsub-neg 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 36.2%

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

      1. mul-1-neg 36.2%

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

      2. distribute-frac-neg 36.2%

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

   8. Simplified 36.2%

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

4. Final simplification 45.5%

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

Alternative 7: 48.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6) (/ 1.0 (/ x s)) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		tmp = 1.0f / (x / s);
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.9999999949504854e-6)) then
        tmp = 1.0e0 / (x / s)
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.9999999949504854e-6))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.9999999949504854e-6))
		tmp = single(1.0) / (x / s);
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.4%

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

      1. mul-1-neg 46.4%

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

      2. unsub-neg 46.4%

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

   5. Simplified 46.4%

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

   6. Taylor expanded in x around inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. distribute-frac-neg 46.4%

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

   8. Simplified 46.4%

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

      1. associate-/r/ 41.7%

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

      2. add-sqr-sqrt 41.7%

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

      3. sqrt-unprod 49.1%

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

      4. sqr-neg 49.1%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. associate-/r/ 46.4%

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

   12. Applied egg-rr 46.4%

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

   if -1.99999999e-6 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 44.9%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.9999999949504854e-6) (/ s x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.9999999949504854e-6f) {
		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 <= (-1.9999999949504854e-6)) 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(-1.9999999949504854e-6))
		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(-1.9999999949504854e-6))
		tmp = s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 46.4%

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

      1. mul-1-neg 46.4%

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

      2. unsub-neg 46.4%

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

   5. Simplified 46.4%

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

   6. Taylor expanded in x around inf 46.4%

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

      1. mul-1-neg 46.4%

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

      2. distribute-frac-neg 46.4%

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

   8. Simplified 46.4%

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

      1. clear-num 41.7%

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

      2. frac-2neg 41.7%

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

      3. remove-double-neg 41.7%

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

      4. expm1-log1p-u 41.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-s}{x}\right)\right)} \]

      5. expm1-udef 97.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-s}{x}\right)} - 1} \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-s} \cdot \sqrt{-s}}}{x}\right)} - 1 \]

      7. sqrt-unprod 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-s\right) \cdot \left(-s\right)}}}{x}\right)} - 1 \]

      8. sqr-neg 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{s \cdot s}}}{x}\right)} - 1 \]

      9. sqrt-unprod 95.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{s} \cdot \sqrt{s}}}{x}\right)} - 1 \]

      10. add-sqr-sqrt 95.9%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{s}}{x}\right)} - 1 \]

   10. Applied egg-rr 95.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{s}{x}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 41.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{s}{x}\right)\right)} \]

       2. expm1-log1p 41.7%

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

   12. Simplified 41.7%

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

   if -1.99999999e-6 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 44.9%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\frac{s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.0% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0 34.6%

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

4. Final simplification 34.6%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))