Logistic function

Percentage Accurate: 99.8% → 99.9%

Time: 14.5s

Alternatives: 13

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))

C

float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

Julia

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

   \[\leadsto e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{-2}\right)}^{\left(\frac{x \cdot 0.5}{s}\right)}} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp -2.0) (/ (* x 0.5) s)))))

C

float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(-2.0f), ((x * 0.5f) / s)));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp((-2.0e0)) ** ((x * 0.5e0) / s)))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(-2.0)) ^ Float32(Float32(x * Float32(0.5)) / s))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(-2.0)) ^ ((x * single(0.5)) / s)));
end

Derivation

1. Initial program 99.8%

   \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
2. Add Preprocessing
3. 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%

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

4. Applied egg-rr 99.8%

   \[\leadsto \frac{1}{1 + \color{blue}{\frac{1}{e^{\frac{x}{s}}}}} \]
5. Step-by-step derivation

   1. inv-pow 99.8%

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

   2. pow-exp 99.8%

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

   3. *-commutative 99.8%

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

   4. pow-exp 99.8%

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

   5. sqr-pow 99.7%

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

   6. pow-prod-down 99.7%

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

   7. prod-exp 99.8%

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

   8. metadata-eval 99.8%

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

   9. div-inv 99.8%

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

   10. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{1}{1 + {\left(e^{-2}\right)}^{\left(\frac{x \cdot 0.5}{s}\right)}} \]
10. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 4: 75.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;t_0 \leq 1:\\ \;\;\;\;0.5 + x \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -5.0)
     (- 1.0 (/ s x))
     (if (<= t_0 1.0) (+ 0.5 (* x (/ 0.25 s))) (/ 1.0 t_0)))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= -5.0f) {
		tmp = 1.0f - (s / x);
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + (x * (0.25f / s));
	} 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 <= (-5.0e0)) then
        tmp = 1.0e0 - (s / x)
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + (x * (0.25e0 / s))
    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(-5.0))
		tmp = Float32(Float32(1.0) - Float32(s / x));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(x * Float32(Float32(0.25) / s)));
	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(-5.0))
		tmp = single(1.0) - (s / x);
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + (x * (single(0.25) / s));
	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) < -5

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. exp-neg 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0 95.9%

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

      1. +-commutative 95.9%

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

   7. Simplified 95.9%

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

   8. Taylor expanded in x around inf 95.8%

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

      1. neg-mul-1 95.8%

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

      2. unsub-neg 95.8%

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

   10. Simplified 95.8%

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

   if -5 < (/.f32 (neg.f32 x) s) < 1

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 95.0%

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

      1. associate-*r/ 95.0%

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

   5. Simplified 95.0%

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

      1. associate-/l* 95.0%

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

      2. associate-/r/ 95.0%

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

   7. Applied egg-rr 95.0%

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

   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 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   5. Simplified 40.9%

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

   6. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   8. Simplified 40.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;1 - \frac{s}{x}\\ \mathbf{elif}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5 + x \cdot \frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0)
     (/ 1.0 (+ 1.0 (/ s x)))
     (if (<= t_0 1.0) (+ 0.5 (* x (/ 0.25 s))) (/ 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 / (1.0f + (s / x));
	} else if (t_0 <= 1.0f) {
		tmp = 0.5f + (x * (0.25f / s));
	} 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 / (1.0e0 + (s / x))
    else if (t_0 <= 1.0e0) then
        tmp = 0.5e0 + (x * (0.25e0 / s))
    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(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	elseif (t_0 <= Float32(1.0))
		tmp = Float32(Float32(0.5) + Float32(x * Float32(Float32(0.25) / s)));
	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) / (single(1.0) + (s / x));
	elseif (t_0 <= single(1.0))
		tmp = single(0.5) + (x * (single(0.25) / s));
	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 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in x around inf 95.3%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 96.3%

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

      1. associate-*r/ 96.3%

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

   5. Simplified 96.3%

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

      1. associate-/l* 96.3%

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

      2. associate-/r/ 96.3%

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

   7. Applied egg-rr 96.3%

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

   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 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   5. Simplified 40.9%

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

   6. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   8. Simplified 40.9%

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

4. Final simplification 74.7%

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

Alternative 6: 72.9% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 -2.0) (- 1.0 (/ s x)) (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 - (s / x);
	} 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 - (s / x)
    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(Float32(1.0) - Float32(s / x));
	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) - (s / x);
	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 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in x around inf 95.2%

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

      1. neg-mul-1 95.2%

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

      2. unsub-neg 95.2%

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

   10. Simplified 95.2%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0 87.6%

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

   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 40.9%

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

      1. mul-1-neg 40.9%

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

      2. unsub-neg 40.9%

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

   5. Simplified 40.9%

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

   6. Taylor expanded in x around inf 40.9%

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

      1. mul-1-neg 40.9%

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

      2. distribute-frac-neg 40.9%

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

   8. Simplified 40.9%

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

4. Final simplification 72.9%

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

Alternative 7: 85.3% accurate, 5.1× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float t_0 = -1.0f + (x / s);
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / (1.0f + (-1.0f / t_0));
	} else {
		tmp = 1.0f / (2.0f + ((x / s) * 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 = (-1.0e0) + (x / s)
    if ((-x / s) <= (-2.0e0)) then
        tmp = 1.0e0 / (1.0e0 + ((-1.0e0) / t_0))
    else
        tmp = 1.0e0 / (2.0e0 + ((x / s) * t_0))
    end if
    code = tmp
end function

Julia

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

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(-1.0) + (x / s);
	tmp = single(0.0);
	if ((-x / s) <= single(-2.0))
		tmp = single(1.0) / (single(1.0) + (single(-1.0) / t_0));
	else
		tmp = single(1.0) / (single(2.0) + ((x / s) * 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. Step-by-step derivation

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

      1. *-un-lft-identity 95.3%

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

      2. frac-2neg 95.3%

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

      3. metadata-eval 95.3%

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

      4. distribute-neg-in 95.3%

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

      5. distribute-frac-neg 95.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 93.1%

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

      8. sqr-neg 93.1%

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

      9. sqrt-unprod 95.6%

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

      10. add-sqr-sqrt 95.6%

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

      11. metadata-eval 95.6%

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

   9. Applied egg-rr 95.6%

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

       1. *-lft-identity 95.6%

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

       2. +-commutative 95.6%

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

   11. Simplified 95.6%

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

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

   1. Initial program 99.7%

      \[\frac{1}{1 + e^{\frac{-x}{s}}} \]
   2. Add Preprocessing
   3. 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%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0 36.1%

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

      1. +-commutative 36.1%

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

   7. Simplified 36.1%

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

   8. Taylor expanded in x around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. unpow2 80.3%

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

      3. unpow2 80.3%

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

      4. times-frac 80.9%

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

      5. distribute-rgt-out 80.9%

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

      6. +-commutative 80.9%

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

   10. Simplified 80.9%

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

4. Final simplification 87.1%

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

Alternative 8: 74.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / (1.0f + (-1.0f / (-1.0f + (x / s))));
	} 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 / (1.0e0 + ((-1.0e0) / ((-1.0e0) + (x / s))))
    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(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(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) / (single(1.0) + (single(-1.0) / (single(-1.0) + (x / s))));
	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 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

      1. *-un-lft-identity 95.3%

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

      2. frac-2neg 95.3%

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

      3. metadata-eval 95.3%

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

      4. distribute-neg-in 95.3%

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

      5. distribute-frac-neg 95.3%

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

      6. add-sqr-sqrt -0.0%

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

      7. sqrt-unprod 93.1%

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

      8. sqr-neg 93.1%

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

      9. sqrt-unprod 95.6%

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

      10. add-sqr-sqrt 95.6%

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

      11. metadata-eval 95.6%

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

   9. Applied egg-rr 95.6%

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

       1. *-lft-identity 95.6%

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

       2. +-commutative 95.6%

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

   11. Simplified 95.6%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 58.7%

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

      1. mul-1-neg 58.7%

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

      2. unsub-neg 58.7%

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

   5. Simplified 58.7%

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

4. Final simplification 74.1%

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

Alternative 9: 67.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (/ (- s) x)
   (if (<= x 3.0000001167615996e-16) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = -s / x;
	} else if (x <= 3.0000001167615996e-16f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = -s / x
    else if (x <= 3.0000001167615996e-16) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(-s) / x);
	elseif (x <= Float32(3.0000001167615996e-16))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = -s / x;
	elseif (x <= single(3.0000001167615996e-16))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.8%

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

      1. mul-1-neg 45.8%

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

      2. unsub-neg 45.8%

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

   5. Simplified 45.8%

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

   6. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   8. Simplified 40.3%

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

   if -1.99999996e-13 < x < 3.0000001e-16

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 3.0000001e-16 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 96.3%

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

      1. +-commutative 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf 94.2%

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

      1. neg-mul-1 94.2%

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

      2. unsub-neg 94.2%

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

   10. Simplified 94.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{elif}\;x \leq 3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13)
   (/ 1.0 (/ x s))
   (if (<= x 3.0000001167615996e-16) 0.5 (- 1.0 (/ s x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		tmp = 1.0f / (x / s);
	} else if (x <= 3.0000001167615996e-16f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f - (s / x);
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-1.99999996490334e-13)) then
        tmp = 1.0e0 / (x / s)
    else if (x <= 3.0000001167615996e-16) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 - (s / x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-1.99999996490334e-13))
		tmp = Float32(Float32(1.0) / Float32(x / s));
	elseif (x <= Float32(3.0000001167615996e-16))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) - Float32(s / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = single(1.0) / (x / s);
	elseif (x <= single(3.0000001167615996e-16))
		tmp = single(0.5);
	else
		tmp = single(1.0) - (s / x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.8%

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

      1. mul-1-neg 45.8%

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

      2. unsub-neg 45.8%

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

   5. Simplified 45.8%

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

   6. Taylor expanded in x around inf 45.8%

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

      1. mul-1-neg 45.8%

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

      2. distribute-frac-neg 45.8%

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

   8. Simplified 45.8%

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

      1. clear-num 40.3%

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

      2. add-sqr-sqrt 40.3%

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

      3. sqrt-unprod 50.3%

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

      4. sqr-neg 50.3%

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

      5. sqrt-unprod -0.0%

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

      6. add-sqr-sqrt 40.2%

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

      7. clear-num 45.7%

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

      8. inv-pow 45.7%

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

   10. Applied egg-rr 45.7%

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

       1. unpow-1 45.7%

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

   12. Simplified 45.7%

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

   if -1.99999996e-13 < x < 3.0000001e-16

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 3.0000001e-16 < x

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 96.3%

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

      1. +-commutative 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in x around inf 94.2%

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

      1. neg-mul-1 94.2%

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

      2. unsub-neg 94.2%

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

   10. Simplified 94.2%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{x}{s}}\\ \mathbf{elif}\;x \leq 3.0000001167615996 \cdot 10^{-16}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{s}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= -2.0f) {
		tmp = 1.0f / (1.0f + (s / x));
	} 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 / (1.0e0 + (s / x))
    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(Float32(1.0) / Float32(Float32(1.0) + Float32(s / x)));
	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) / (single(1.0) + (s / x));
	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 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. exp-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in x around inf 95.3%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 58.7%

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

      1. mul-1-neg 58.7%

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

      2. unsub-neg 58.7%

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

   5. Simplified 58.7%

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

4. Final simplification 74.0%

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

Alternative 12: 46.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x -1.99999996490334e-13) (/ (- s) x) 0.5))

C

float code(float x, float s) {
	float tmp;
	if (x <= -1.99999996490334e-13f) {
		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 <= (-1.99999996490334e-13)) 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(-1.99999996490334e-13))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-1.99999996490334e-13))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -1.99999996e-13

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 45.8%

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

      1. mul-1-neg 45.8%

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

      2. unsub-neg 45.8%

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

   5. Simplified 45.8%

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

   6. Taylor expanded in x around inf 40.3%

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

      1. associate-*r/ 40.3%

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

      2. neg-mul-1 40.3%

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

   8. Simplified 40.3%

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

   if -1.99999996e-13 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 44.3%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.99999996490334 \cdot 10^{-13}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.0% 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 99.8%

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

3. Taylor expanded in x around 0 31.7%

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

4. Final simplification 31.7%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024023 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))