Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 1.5s

Alternatives: 8

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 45.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 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 + exp((-x / s))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.2%

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

Alternative 3: 44.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;1 + \frac{1}{\frac{1}{-x}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 0.4000000059604645)
   (+ 1.0 (/ 1.0 (/ 1.0 (- x))))
   (+ 1.0 (exp (- x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = 1.0f + (1.0f / (1.0f / -x));
	} else {
		tmp = 1.0f + expf(-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 <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (1.0e0 / (1.0e0 / -x))
    else
        tmp = 1.0e0 + exp(-x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) / Float32(-x))));
	else
		tmp = Float32(Float32(1.0) + exp(Float32(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.4000000059604645))
		tmp = single(1.0) + (single(1.0) / (single(1.0) / -x));
	else
		tmp = single(1.0) + exp(-x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \left(1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}\right) \]

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.0%

       \[\leadsto 1 + \frac{1}{\frac{1}{-x}} \]

   if 0.400000006 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

      \[\leadsto 1 + e^{-x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 43.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{-x}\\ \mathbf{if}\;x \leq 0.4000000059604645:\\ \;\;\;\;1 + \frac{1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ 1.0 (- x))))
   (if (<= x 0.4000000059604645) (+ 1.0 (/ 1.0 t_0)) (+ 1.0 t_0))))

C

float code(float x, float s) {
	float t_0 = 1.0f / -x;
	float tmp;
	if (x <= 0.4000000059604645f) {
		tmp = 1.0f + (1.0f / t_0);
	} 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 = 1.0e0 / -x
    if (x <= 0.4000000059604645e0) then
        tmp = 1.0e0 + (1.0e0 / t_0)
    else
        tmp = 1.0e0 + t_0
    end if
    code = tmp
end function

Julia

function code(x, s)
	t_0 = Float32(Float32(1.0) / Float32(-x))
	tmp = Float32(0.0)
	if (x <= Float32(0.4000000059604645))
		tmp = Float32(Float32(1.0) + Float32(Float32(1.0) / t_0));
	else
		tmp = Float32(Float32(1.0) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	t_0 = single(1.0) / -x;
	tmp = single(0.0);
	if (x <= single(0.4000000059604645))
		tmp = single(1.0) + (single(1.0) / t_0);
	else
		tmp = single(1.0) + t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 0.400000006

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \left(1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}\right) \]

   8. Applied rewrites 5.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 28.0%

       \[\leadsto 1 + \frac{1}{\frac{1}{-x}} \]

   if 0.400000006 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \left(1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}\right) \]

   8. Applied rewrites 96.4%

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

Alternative 5: 32.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.0000000116860974 \cdot 10^{-7}:\\ \;\;\;\;1 + \frac{-x}{s}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{1}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= x 1.0000000116860974e-7) (+ 1.0 (/ (- x) s)) (+ 1.0 (/ 1.0 (- x)))))

C

float code(float x, float s) {
	float tmp;
	if (x <= 1.0000000116860974e-7f) {
		tmp = 1.0f + (-x / s);
	} else {
		tmp = 1.0f + (1.0f / -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 <= 1.0000000116860974e-7) then
        tmp = 1.0e0 + (-x / s)
    else
        tmp = 1.0e0 + (1.0e0 / -x)
    end if
    code = tmp
end function

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(1.0000000116860974e-7))
		tmp = Float32(Float32(1.0) + Float32(Float32(-x) / s));
	else
		tmp = Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(-x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(1.0000000116860974e-7))
		tmp = single(1.0) + (-x / s);
	else
		tmp = single(1.0) + (single(1.0) / -x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < 1.00000001e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 12.2%

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

   if 1.00000001e-7 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 + \left(1 + \color{blue}{x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}\right) \]

   8. Applied rewrites 79.0%

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

Alternative 6: 9.3% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 9.3%

   \[\leadsto 1 + \frac{-x}{\color{blue}{s}} \]
9. Add Preprocessing

Alternative 7: 6.1% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-x} \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{2}} \]

5. Applied rewrites 6.1%

   \[\leadsto \frac{1}{\color{blue}{-x}} \]
6. Add Preprocessing

Alternative 8: 4.9% accurate, 42.7× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x s) :precision binary32 (- x))

C

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

Fortran

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

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = -x;
end

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} + x \cdot \left(\frac{-1}{48} \cdot \frac{{x}^{2}}{{s}^{3}} + \frac{1}{4} \cdot \frac{1}{s}\right)} \]

5. Applied rewrites 10.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{480} \cdot \frac{{x}^{2}}{{s}^{5}} - \frac{1}{48} \cdot \frac{1}{{s}^{3}}\right) + \frac{1}{4} \cdot \frac{1}{s}\right)} \]

8. Applied rewrites 5.0%

   \[\leadsto \color{blue}{-x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))