Logistic function

Percentage Accurate: 99.8% → 99.8%
Time: 9.5s
Alternatives: 3
Speedup: 1.0×

Specification

?
\[0 \leq s \land s \leq 1.0651631\]
\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
(FPCore (x s)
  :precision binary32
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

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

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

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

\frac{1}{1 + e^{\frac{-x}{s}}}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 3 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.8% accurate, 1.0× speedup?

\[\frac{1}{1 + e^{\frac{-x}{s}}} \]
(FPCore (x s)
  :precision binary32
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))
float code(float x, float s) {
	return 1.0f / (1.0f + expf((-x / s)));
}

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

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

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

\frac{1}{1 + e^{\frac{-x}{s}}}

Alternative 1: 69.0% accurate, 1.0× speedup?

\[\begin{array}{l} \mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 4000:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
(FPCore (x s)
  :precision binary32
  (if (<= (+ 1.0 (exp (/ (- x) s))) 4000.0) 0.5 0.0))
float code(float x, float s) {
	float tmp;
	if ((1.0f + expf((-x / s))) <= 4000.0f) {
		tmp = 0.5f;
	} else {
		tmp = 0.0f;
	}
	return tmp;
}

real(4) function code(x, s)
use fmin_fmax_functions
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if ((1.0e0 + exp((-x / s))) <= 4000.0e0) then
        tmp = 0.5e0
    else
        tmp = 0.0e0
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) + exp(Float32(Float32(-x) / s))) <= Float32(4000.0))
		tmp = Float32(0.5);
	else
		tmp = Float32(0.0);
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if ((single(1.0) + exp((-x / s))) <= single(4000.0))
		tmp = single(0.5);
	else
		tmp = single(0.0);
	end
	tmp_2 = tmp;
end

\begin{array}{l}
\mathbf{if}\;1 + e^{\frac{-x}{s}} \leq 4000:\\
\;\;\;\;0.5\\

\mathbf{else}:\\
\;\;\;\;0\\


\end{array}
Derivation
  1. Split input into 2 regimes

  2. if (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s))) < 4e3

    1. Initial program 99.8%

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

    2. Taylor expanded in x around 0

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

      1. Applied rewrites34.9%

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

      if 4e3 < (+.f32 #s(literal 1 binary32) (exp.f32 (/.f32 (neg.f32 x) s)))

      1. Initial program 99.8%

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

      2. Taylor expanded in x around 0

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

        1. Applied rewrites34.9%

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

        2. Taylor expanded in undef-var around zero

          \[\leadsto 0 \]
        3. Step-by-step derivation

          1. Applied rewrites40.4%

            \[\leadsto 0 \]
        4. Recombined 2 regimes into one program.
        5. Add Preprocessing

        Alternative 2: 34.9% accurate, 23.2× speedup?

        \[0.5 \]
        (FPCore (x s)
  :precision binary32
  0.5)
        float code(float x, float s) {
	return 0.5f;
}

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

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

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

        0.5
        Derivation

        1. Initial program 99.8%

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

        2. Taylor expanded in x around 0

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

          1. Applied rewrites34.9%

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

          Reproduce

          ?
          herbie shell --seed 2025322 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))