Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 9.9s

Alternatives: 15

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(\sqrt{e^{-1}}\right)}^{\left(\frac{x}{s} \cdot 2\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (sqrt (exp -1.0)) (* (/ x s) 2.0)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(sqrtf(expf(-1.0f)), ((x / s) * 2.0f)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (sqrt(exp((-1.0e0))) ** ((x / s) * 2.0e0)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (sqrt(exp(Float32(-1.0))) ^ Float32(Float32(x / s) * Float32(2.0)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (sqrt(exp(single(-1.0))) ^ ((x / s) * single(2.0))));
end

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.8%

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

   2. exp-prod 84.5%

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

   3. neg-mul-1 84.5%

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

   4. exp-prod 84.5%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.8%

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

   2. unpow-prod-down 99.8%

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

5. Applied egg-rr 99.8%

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

   1. pow-sqr 99.8%

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

   2. count-2 99.8%

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

   3. *-rgt-identity 99.8%

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

   4. *-rgt-identity 99.8%

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

   5. distribute-lft-out 99.8%

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

   6. metadata-eval 99.8%

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

7. Simplified 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. div-inv 99.8%

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

   2. exp-prod 84.5%

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

   3. neg-mul-1 84.5%

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

   4. exp-prod 84.5%

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

   5. pow-pow 99.8%

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

   6. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 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 99.7%

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

2. Final simplification 99.7%

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

Alternative 4: 89.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq 2.999999997758287 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(x \cdot \left(x \cdot \frac{\frac{1}{s}}{s}\right)\right) - \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (- x) 2.999999997758287e-32)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (- (* 0.5 (* x (* x (/ (/ 1.0 s) s)))) (/ x s))))))

C

float code(float x, float s) {
	float tmp;
	if (-x <= 2.999999997758287e-32f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + ((0.5f * (x * (x * ((1.0f / s) / s)))) - (x / s)));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= 2.999999997758287e-32) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x * (x * ((1.0e0 / s) / s)))) - (x / s)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(2.999999997758287e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x * Float32(x * Float32(Float32(Float32(1.0) / s) / s)))) - Float32(x / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(2.999999997758287e-32))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x * (x * ((single(1.0) / s) / s)))) - (x / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (neg.f32 x) < 3e-32

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 93.9%

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

   if 3e-32 < (neg.f32 x)

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. unsub-neg 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 78.1%

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

   4. Simplified 78.1%

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

      1. clear-num 78.1%

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

      2. frac-times 82.2%

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

      3. *-un-lft-identity 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. div-inv 85.9%

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

      2. *-commutative 85.9%

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

   8. Applied egg-rr 85.9%

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

      1. associate-/r* 85.9%

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

      2. associate-/r/ 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-x \leq 2.999999997758287 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \left(x \cdot \left(x \cdot \frac{\frac{1}{s}}{s}\right)\right) - \frac{x}{s}\right)}\\ \end{array} \]

Alternative 5: 86.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 1.0) (+ 0.5 (* (/ x s) 0.25)) (* 2.0 (/ s (* x (/ x s))))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} else {
		tmp = 2.0f * (s / (x * (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 = 1.0e0
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    else
        tmp = 2.0e0 * (s / (x * (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(1.0);
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	else
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * 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(1.0);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(2.0) * (s / (x * (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 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in s around 0 98.2%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.3%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. unsub-neg 82.8%

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

      3. unpow2 82.8%

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

      4. unpow2 82.8%

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

      5. times-frac 73.9%

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

   4. Simplified 73.9%

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

   5. Taylor expanded in x around inf 81.8%

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

      1. unpow2 81.8%

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

      2. associate-/l* 71.2%

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

      3. unpow2 71.2%

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

      4. associate-*l/ 71.5%

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

      5. *-commutative 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{s}{x \cdot \frac{x}{s}}\\ \end{array} \]

Alternative 6: 89.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.999999997758287 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -2.999999997758287e-32)
   (/ 1.0 (+ 2.0 (- (* 0.5 (/ x (/ (* s s) x))) (/ x s))))
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -2.999999997758287e-32f) {
		tmp = 1.0f / (2.0f + ((0.5f * (x / ((s * s) / x))) - (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 <= (-2.999999997758287e-32)) then
        tmp = 1.0e0 / (2.0e0 + ((0.5e0 * (x / ((s * s) / x))) - (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(-2.999999997758287e-32))
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(Float32(0.5) * Float32(x / Float32(Float32(s * s) / x))) - 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(-2.999999997758287e-32))
		tmp = single(1.0) / (single(2.0) + ((single(0.5) * (x / ((s * s) / x))) - (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 2 regimes

2. if x < -3e-32

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 85.3%

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

      1. mul-1-neg 85.3%

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

      2. unsub-neg 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 78.1%

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

   4. Simplified 78.1%

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

      1. clear-num 78.1%

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

      2. frac-times 82.2%

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

      3. *-un-lft-identity 82.2%

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

   6. Applied egg-rr 82.2%

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

      1. associate-*l/ 86.3%

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

   8. Applied egg-rr 86.3%

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

   if -3e-32 < x

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 93.9%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.999999997758287 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{2 + \left(0.5 \cdot \frac{x}{\frac{s \cdot s}{x}} - \frac{x}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \end{array} \]

Alternative 7: 85.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 0.10000000149011612)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (* 0.5 (* (/ x s) (/ x s)))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 0.10000000149011612f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + (0.5f * ((x / s) * (x / s))));
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((-x / s) <= 0.10000000149011612e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * ((x / s) * (x / s))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(0.10000000149011612))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(Float32(x / s) * Float32(x / s)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(0.10000000149011612))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + (single(0.5) * ((x / s) * (x / s))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 95.1%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 81.2%

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

      1. mul-1-neg 81.2%

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

      2. unsub-neg 81.2%

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

      3. unpow2 81.2%

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

      4. unpow2 81.2%

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

      5. times-frac 73.3%

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

   4. Simplified 73.3%

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

      1. clear-num 73.3%

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

      2. frac-times 78.2%

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

      3. *-un-lft-identity 78.2%

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

   6. Applied egg-rr 78.2%

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

   7. Taylor expanded in x around inf 81.2%

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

      1. unpow2 81.2%

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

      2. unpow2 81.2%

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

   9. Simplified 81.2%

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

       1. times-frac 73.1%

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

   11. Applied egg-rr 73.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 0.10000000149011612:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(\frac{x}{s} \cdot \frac{x}{s}\right)}\\ \end{array} \]

Alternative 8: 89.7% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 20.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ x s)))))
   (/ 1.0 (+ 2.0 (* 0.5 (* x (/ x (* s s))))))))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 20.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 1.0f / (2.0f + (0.5f * (x * (x / (s * 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) <= 20.0e0) then
        tmp = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 1.0e0 / (2.0e0 + (0.5e0 * (x * (x / (s * s)))))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(20.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(0.5) * Float32(x * Float32(x / Float32(s * s))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((-x / s) <= single(20.0))
		tmp = single(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(1.0) / (single(2.0) + (single(0.5) * (x * (x / (s * s)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (/.f32 (neg.f32 x) s) < 20

   1. Initial program 99.7%

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

      1. distribute-frac-neg 99.7%

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

      2. exp-neg 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 92.3%

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

   if 20 < (/.f32 (neg.f32 x) s)

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 85.9%

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

      1. mul-1-neg 85.9%

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

      2. unsub-neg 85.9%

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

      3. unpow2 85.9%

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

      4. unpow2 85.9%

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

      5. times-frac 76.2%

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

   4. Simplified 76.2%

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

      1. clear-num 76.2%

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

      2. frac-times 81.3%

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

      3. *-un-lft-identity 81.3%

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

   6. Applied egg-rr 81.3%

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

   7. Taylor expanded in x around inf 85.9%

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

      1. unpow2 85.9%

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

      2. unpow2 85.9%

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

   9. Simplified 85.9%

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

       1. associate-/l* 86.9%

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

       2. associate-/r/ 86.9%

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

   11. Applied egg-rr 86.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 20:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{x}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 + 0.5 \cdot \left(x \cdot \frac{x}{s \cdot s}\right)}\\ \end{array} \]

Alternative 9: 76.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     1.0
     (if (<= t_0 1.0) (+ 0.5 (* (/ x s) 0.25)) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -2.0f) {
		tmp = 1.0f;
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + ((x / s) * 0.25f);
	} 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 = 1.0e0
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + ((x / s) * 0.25e0)
    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(1.0);
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(Float32(x / s) * Float32(0.25)));
	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(1.0);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + ((x / s) * single(0.25));
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in s around 0 98.2%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 96.3%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot \frac{x}{s}} \]
   3. Step-by-step derivation

      1. *-commutative 96.3%

         \[\leadsto 0.5 + \color{blue}{\frac{x}{s} \cdot 0.25} \]

   4. Simplified 96.3%

      \[\leadsto \color{blue}{0.5 + \frac{x}{s} \cdot 0.25} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 48.5%

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

      1. mul-1-neg 48.5%

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

      2. unsub-neg 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in x around inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. distribute-frac-neg 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -2:\\ \;\;\;\;1\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + \frac{x}{s} \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]

Alternative 10: 74.3% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0) 1.0 (if (<= t_0 1.0) 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 = 1.0f;
	} else if (t_0 <= 1.0f) {
		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 = 1.0e0
    else if (t_0 <= 1.0e0) 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(1.0);
	elseif (t_0 <= Float32(1.0))
		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(1.0);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5);
	else
		tmp = single(1.0) / t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in s around 0 98.2%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 88.0%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 48.5%

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

      1. mul-1-neg 48.5%

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

      2. unsub-neg 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in x around inf 48.5%

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

      1. mul-1-neg 48.5%

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

      2. distribute-frac-neg 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 76.1%

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

Alternative 11: 85.0% accurate, 6.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (x / s))));
	} else {
		tmp = 2.0f * (s / (x * (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 = 1.0e0 / (1.0e0 + (1.0e0 / (1.0e0 + (x / s))))
    else
        tmp = 2.0e0 * (s / (x * (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(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(x / s)))));
	else
		tmp = Float32(Float32(2.0) * Float32(s / Float32(x * 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(1.0) / (single(1.0) + (single(1.0) / (single(1.0) + (x / s))));
	else
		tmp = single(2.0) * (s / (x * (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 99.8%

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

      1. distribute-frac-neg 99.8%

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

      2. exp-neg 99.8%

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

   3. Applied egg-rr 99.8%

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

   4. Taylor expanded in x around 0 94.3%

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

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 82.8%

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

      1. mul-1-neg 82.8%

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

      2. unsub-neg 82.8%

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

      3. unpow2 82.8%

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

      4. unpow2 82.8%

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

      5. times-frac 73.9%

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

   4. Simplified 73.9%

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

   5. Taylor expanded in x around inf 81.8%

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

      1. unpow2 81.8%

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

      2. associate-/l* 71.2%

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

      3. unpow2 71.2%

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

      4. associate-*l/ 71.5%

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

      5. *-commutative 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 85.4%

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

Alternative 12: 75.7% accurate, 8.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f;
	} 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) <= (-2.0e0)) then
        tmp = 1.0e0
    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(-2.0))
		tmp = Float32(1.0);
	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(-2.0))
		tmp = single(1.0);
	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) < -2

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 94.9%

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

   5. Taylor expanded in x around inf 94.8%

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

      1. +-commutative 94.8%

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

      2. mul-1-neg 94.8%

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

      3. unsub-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in s around 0 98.2%

      \[\leadsto \color{blue}{1} \]

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

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 66.6%

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

      1. mul-1-neg 66.6%

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

      2. unsub-neg 66.6%

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

   4. Simplified 66.6%

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

4. Final simplification 77.6%

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

Alternative 13: 69.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -0.0010000000474974513)
   (/ (- s) x)
   (if (<= x 4.999999841327613e-21) 0.5 1.0)))

C

float code(float x, float s) {
	float tmp;
	if (x <= -0.0010000000474974513f) {
		tmp = -s / x;
	} else if (x <= 4.999999841327613e-21f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-0.0010000000474974513e0)) then
        tmp = -s / x
    else if (x <= 4.999999841327613e-21) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-0.0010000000474974513))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(4.999999841327613e-21))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-0.0010000000474974513))
		tmp = -s / x;
	elseif (x <= single(4.999999841327613e-21))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.00100000005

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 61.1%

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

      1. mul-1-neg 61.1%

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

      2. unsub-neg 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in x around inf 55.0%

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

      1. associate-*r/ 55.0%

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

      2. neg-mul-1 55.0%

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

   7. Simplified 55.0%

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

   if -0.00100000005 < x < 4.99999984e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0 63.4%

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

   if 4.99999984e-21 < x

   1. Initial program 99.9%

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

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 95.3%

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

   5. Taylor expanded in x around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in s around 0 93.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0010000000474974513:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 14: 57.8% accurate, 34.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (if (<= x 4.999999841327613e-21) 0.5 1.0))

C

float code(float x, float s) {
	float tmp;
	if (x <= 4.999999841327613e-21f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 4.999999841327613e-21) then
        tmp = 0.5e0
    else
        tmp = 1.0e0
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(4.999999841327613e-21))
		tmp = Float32(0.5);
	else
		tmp = Float32(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(4.999999841327613e-21))
		tmp = single(0.5);
	else
		tmp = single(1.0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 4.99999984e-21

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 38.0%

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

   if 4.99999984e-21 < x

   1. Initial program 99.9%

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

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 95.3%

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

   5. Taylor expanded in x around inf 90.0%

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

      1. +-commutative 90.0%

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

      2. mul-1-neg 90.0%

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

      3. unsub-neg 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in s around 0 93.2%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.999999841327613 \cdot 10^{-21}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 34.9% accurate, 108.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 35.5%

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

3. Final simplification 35.5%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023218 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))