Logistic distribution

Percentage Accurate: 99.5% → 99.5%

Time: 12.2s

Alternatives: 11

Speedup: 2.9×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))

C

float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

Julia

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

MATLAB

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

Alternative 1: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{2 + 2 \cdot \cosh \left(\frac{\left|x\right|}{s}\right)}}{s} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 (+ 2.0 (* 2.0 (cosh (/ (fabs x) s))))) s))

C

float code(float x, float s) {
	return (1.0f / (2.0f + (2.0f * coshf((fabsf(x) / s))))) / s;
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(Float32(1.0) / Float32(Float32(2.0) + Float32(Float32(2.0) * cosh(Float32(abs(x) / s))))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(1.0) / (single(2.0) + (single(2.0) * cosh((abs(x) / s))))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(2 + \cosh \left(\frac{\left|x\right|}{s}\right) \cdot 2\right)} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (+ 2.0 (* (cosh (/ (fabs x) s)) 2.0)))))

C

float code(float x, float s) {
	return 1.0f / (s * (2.0f + (coshf((fabsf(x) / s)) * 2.0f)));
}

Fortran

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

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(2.0) + Float32(cosh(Float32(abs(x) / s)) * Float32(2.0)))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * (single(2.0) + (cosh((abs(x) / s)) * single(2.0))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 3: 94.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{e^{\frac{\left|x\right|}{s}}}}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ (/ 0.25 (exp (/ (fabs x) s))) s))

C

float code(float x, float s) {
	return (0.25f / expf((fabsf(x) / s))) / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (0.25e0 / exp((abs(x) / s))) / s
end function

Julia

function code(x, s)
	return Float32(Float32(Float32(0.25) / exp(Float32(abs(x) / s))) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = (single(0.25) / exp((abs(x) / s))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in x around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 4: 54.6% accurate, 34.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.99999999855967 \cdot 10^{-23}:\\ \;\;\;\;\frac{0.25 + \frac{-0.0625}{\frac{\frac{s}{x}}{\frac{x}{s}}}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{x \cdot x}{s \cdot s} + 4}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 7.99999999855967e-23)
   (/ (+ 0.25 (/ -0.0625 (/ (/ s x) (/ x s)))) s)
   (/ (/ 1.0 (+ (/ (* x x) (* s s)) 4.0)) s)))

C

float code(float x, float s) {
	float tmp;
	if (x <= 7.99999999855967e-23f) {
		tmp = (0.25f + (-0.0625f / ((s / x) / (x / s)))) / s;
	} else {
		tmp = (1.0f / (((x * x) / (s * s)) + 4.0f)) / s;
	}
	return tmp;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 7.99999999855967e-23) then
        tmp = (0.25e0 + ((-0.0625e0) / ((s / x) / (x / s)))) / s
    else
        tmp = (1.0e0 / (((x * x) / (s * s)) + 4.0e0)) / s
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(7.99999999855967e-23))
		tmp = Float32(Float32(Float32(0.25) + Float32(Float32(-0.0625) / Float32(Float32(s / x) / Float32(x / s)))) / s);
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(Float32(Float32(x * x) / Float32(s * s)) + Float32(4.0))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(7.99999999855967e-23))
		tmp = (single(0.25) + (single(-0.0625) / ((s / x) / (x / s)))) / s;
	else
		tmp = (single(1.0) / (((x * x) / (s * s)) + single(4.0))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 8e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 8e-23 < x

   1. Initial program 99.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   6. Applied egg-rr 0

      \[\leadsto expr\]

   7. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   9. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 45.9% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot \left(x \cdot x\right)}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026) (/ 0.25 s) (/ 1.0 (/ (* 3.0 (* x x)) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / ((3.0f * (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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / ((3.0e0 * (x * x)) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(3.0) * Float32(x * x)) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / ((single(3.0) * (x * x)) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.00200000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   12. Taylor expanded in x around inf 0

       \[\leadsto expr\]

   14. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 45.9% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{3 \cdot \frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026) (/ 0.25 s) (/ 1.0 (* 3.0 (/ (* x x) s)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = 1.0f / (3.0f * ((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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = 1.0e0 / (3.0e0 * ((x * x) / s))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(3.0) * Float32(Float32(x * x) / s)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = single(1.0) / (single(3.0) * ((x * x) / s));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.00200000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 45.8% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{1}{\frac{x}{\frac{s}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026)
   (/ 0.25 s)
   (* 0.3333333333333333 (/ 1.0 (/ x (/ s x))))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f * (1.0f / (x / (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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 * (1.0e0 / (x / (s / x)))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) * Float32(Float32(1.0) / Float32(x / Float32(s / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) * (single(1.0) / (x / (s / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.00200000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.4% accurate, 47.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\frac{\frac{x \cdot x}{s}}{s} + 4\right)} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 1.0 (* s (+ (/ (/ (* x x) s) s) 4.0))))

C

float code(float x, float s) {
	return 1.0f / (s * ((((x * x) / s) / s) + 4.0f));
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (s * ((((x * x) / s) / s) + 4.0e0))
end function

Julia

function code(x, s)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(Float32(Float32(x * x) / s) / s) + Float32(4.0))))
end

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (s * ((((x * x) / s) / s) + single(4.0)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]

8. Applied egg-rr 0

   \[\leadsto expr\]

9. Taylor expanded in s around inf 0

   \[\leadsto expr\]

11. Simplified 0

    \[\leadsto expr\]
12. Add Preprocessing

Alternative 9: 45.8% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{x \cdot x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026)
   (/ 0.25 s)
   (/ 0.3333333333333333 (/ (* x x) s))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f / ((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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 / ((x * x) / s)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) / Float32(Float32(x * x) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) / ((x * x) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.00200000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]

   13. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 45.2% accurate, 51.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0020000000949949026:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{s}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.0020000000949949026)
   (/ 0.25 s)
   (* 0.3333333333333333 (/ s (* x x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.0020000000949949026f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f * (s / (x * 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 <= 0.0020000000949949026e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 * (s / (x * x))
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) * Float32(s / Float32(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0020000000949949026))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) * (s / (x * x));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.00200000009

   1. Initial program 99.6%

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

   3. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 0.00200000009 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in s around inf 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]

   9. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   11. Simplified 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 26.7% accurate, 206.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (/ 0.25 s))

C

float code(float x, float s) {
	return 0.25f / s;
}

Fortran

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

Julia

function code(x, s)
	return Float32(Float32(0.25) / s)
end

MATLAB

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))