Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 8.7s

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

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

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 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 60.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x}{s}}{s}\\ \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(t\_0, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(0.5 \cdot t\_0\right) \cdot x + 2\right) - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (/ x s) s)))
   (if (<= (/ (- x) s) -5.0)
     (/
      1.0
      (+
       (fma (fma t_0 (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s)) x 1.0)
       1.0))
     (/ 1.0 (- (+ (* (* 0.5 t_0) x) 2.0) (/ x s))))))

C

float code(float x, float s) {
	float t_0 = (x / s) / s;
	float tmp;
	if ((-x / s) <= -5.0f) {
		tmp = 1.0f / (fmaf(fmaf(t_0, fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), x, 1.0f) + 1.0f);
	} else {
		tmp = 1.0f / ((((0.5f * t_0) * x) + 2.0f) - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(x / s) / s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(-5.0))
		tmp = Float32(Float32(1.0) / Float32(fma(fma(t_0, fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), x, Float32(1.0)) + Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(Float32(Float32(Float32(0.5) * t_0) * x) + Float32(2.0)) - Float32(x / s)));
	end
	return tmp
end

Derivation

1. Split input into 2 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. Taylor expanded in s around inf

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

   5. Applied rewrites 28.9%

      \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right)}} \]

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 37.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 83.8%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq -5:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{\frac{x}{s}}{s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), x, 1\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\left(0.5 \cdot \frac{\frac{x}{s}}{s}\right) \cdot x + 2\right) - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 84.5%

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

Alternative 4: 67.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{s}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;t\_0 \leq 9.999999616903162 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{s}, 0\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (- x) s)))
   (if (<= t_0 1.0)
     0.5
     (if (<= t_0 9.999999616903162e+35)
       (* (fma x (/ 1.0 s) 0.0) 0.25)
       (/ 1.0 (- 2.0 (/ x s)))))))

C

float code(float x, float s) {
	float t_0 = -x / s;
	float tmp;
	if (t_0 <= 1.0f) {
		tmp = 0.5f;
	} else if (t_0 <= 9.999999616903162e+35f) {
		tmp = fmaf(x, (1.0f / s), 0.0f) * 0.25f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

Julia

function code(x, s)
	t_0 = Float32(Float32(-x) / s)
	tmp = Float32(0.0)
	if (t_0 <= Float32(1.0))
		tmp = Float32(0.5);
	elseif (t_0 <= Float32(9.999999616903162e+35))
		tmp = Float32(fma(x, Float32(Float32(1.0) / s), Float32(0.0)) * Float32(0.25));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.7%

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

   if 1 < (/.f32 (neg.f32 x) s) < 9.99999962e35

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      5. lower-/.f32 6.3

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   7. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 4.6%

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

   12. Applied rewrites 100.0%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(x, \frac{1}{\color{blue}{s}}, 0\right) \]

   if 9.99999962e35 < (/.f32 (neg.f32 x) s)

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-/.f32 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{elif}\;\frac{-x}{s} \leq 9.999999616903162 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{s}, 0\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.7%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in s around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

   5. Applied rewrites 6.3%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{x}{s}, \mathsf{fma}\left(\frac{0.5}{s}, x, -1\right), 2\right)}} \]

   6. Taylor expanded in s around 0

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

   8. Applied rewrites 84.5%

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

Alternative 6: 57.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{s}, 0\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1.0) 0.5 (* (fma x (/ 1.0 s) 0.0) 0.25)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = fmaf(x, (1.0f / s), 0.0f) * 0.25f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(fma(x, Float32(Float32(1.0) / s), Float32(0.0)) * Float32(0.25));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.7%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      5. lower-/.f32 6.3

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   7. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 4.1%

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

   12. Applied rewrites 100.0%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(x, \frac{1}{\color{blue}{s}}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{s}, 0\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{s}, 0\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (/ (- x) s) 1.0) 0.5 (* (fma 1.0 (/ x s) 0.0) 0.25)))

C

float code(float x, float s) {
	float tmp;
	if ((-x / s) <= 1.0f) {
		tmp = 0.5f;
	} else {
		tmp = fmaf(1.0f, (x / s), 0.0f) * 0.25f;
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(-x) / s) <= Float32(1.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(fma(Float32(1.0), Float32(x / s), Float32(0.0)) * Float32(0.25));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

   3. Taylor expanded in s around inf

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

   5. Applied rewrites 50.7%

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

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

   1. Initial program 100.0%

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

      1. lift-/.f32 N/A

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

      2. inv-pow N/A

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

      3. sqr-pow N/A

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

      4. pow2 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in s around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. lower-fma.f32 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{4}, \frac{x}{s}, \frac{1}{2}\right)} \]

      5. lower-/.f32 6.3

         \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{\frac{x}{s}}, 0.5\right) \]

   7. Applied rewrites 6.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.25, \frac{x}{s}, 0.5\right)} \]

   8. Taylor expanded in s around 0

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

   10. Applied rewrites 4.1%

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

   12. Applied rewrites 100.0%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(1, \frac{x}{\color{blue}{s}}, 0\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-x}{s} \leq 1:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{s}, 0\right) \cdot 0.25\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.5% accurate, 128.0× speedup

Math

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

FPCore

(FPCore (x s) :precision binary32 0.5)

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 33.9%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))