Logistic function

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

float code(float x, float s) {
	return 1.0f / (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 + (1.0e0 / exp((x / s))))
end function

Julia

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

MATLAB

function tmp = code(x, s)
	tmp = single(1.0) / (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. Step-by-step derivation

   1. lift-exp.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. exp-neg N/A

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

   6. lower-/.f32 N/A

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

   7. lower-exp.f32 N/A

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

   8. lower-/.f32 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{\frac{x}{-s}} \leq 39999999311872:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x s)
 :precision binary32
 (if (<= (exp (/ x (- s))) 39999999311872.0)
   (/ 1.0 (+ 1.0 (/ 1.0 (+ 1.0 (/ (fma x (* (/ x s) 0.5) x) s)))))
   (/
    1.0
    (fma
     x
     (fma (/ x (* s s)) (fma -0.16666666666666666 (/ x s) 0.5) (/ -1.0 s))
     2.0))))

C

float code(float x, float s) {
	float tmp;
	if (expf((x / -s)) <= 39999999311872.0f) {
		tmp = 1.0f / (1.0f + (1.0f / (1.0f + (fmaf(x, ((x / s) * 0.5f), x) / s))));
	} else {
		tmp = 1.0f / fmaf(x, fmaf((x / (s * s)), fmaf(-0.16666666666666666f, (x / s), 0.5f), (-1.0f / s)), 2.0f);
	}
	return tmp;
}

Julia

function code(x, s)
	tmp = Float32(0.0)
	if (exp(Float32(x / Float32(-s))) <= Float32(39999999311872.0))
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(fma(x, Float32(Float32(x / s) * Float32(0.5)), x) / s)))));
	else
		tmp = Float32(Float32(1.0) / fma(x, fma(Float32(x / Float32(s * s)), fma(Float32(-0.16666666666666666), Float32(x / s), Float32(0.5)), Float32(Float32(-1.0) / s)), Float32(2.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (exp.f32 (/.f32 (neg.f32 x) s)) < 3.99999993e13

   1. Initial program 99.8%

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

      1. lift-exp.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. distribute-frac-neg N/A

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

      5. exp-neg N/A

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

      6. lower-/.f32 N/A

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

      7. lower-exp.f32 N/A

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

      8. lower-/.f32 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in s around -inf

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

      1. lower-+.f32 N/A

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

      2. associate-*r/ N/A

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

      3. mul-1-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-neg-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. lower-/.f32 N/A

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

   7. Applied rewrites 96.0%

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

   if 3.99999993e13 < (exp.f32 (/.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

      \[\leadsto \frac{1}{\color{blue}{2 + x \cdot \left(x \cdot \left(\frac{-1}{6} \cdot \frac{x}{{s}^{3}} + \frac{1}{2} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{s}\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 88.5%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\frac{x}{-s}} \leq 39999999311872:\\ \;\;\;\;\frac{1}{1 + \frac{1}{1 + \frac{\mathsf{fma}\left(x, \frac{x}{s} \cdot 0.5, x\right)}{s}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{x}{s \cdot s}, \mathsf{fma}\left(-0.16666666666666666, \frac{x}{s}, 0.5\right), \frac{-1}{s}\right), 2\right)}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))