Logistic function

Percentage Accurate: 99.8% → 99.9%
Time: 16.3s
Alternatives: 8
Speedup: 1.0×

Specification

?
\[0 \leq s \land s \leq 1.0651631\]
\[\begin{array}{l} \\ \frac{1}{1 + e^{\frac{-x}{s}}} \end{array} \]
(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)
    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

\begin{array}{l}

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

Sampling outcomes in binary32 precision:

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 8 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?

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

\begin{array}{l}

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

Alternative 1: 99.9% accurate, 0.4× speedup?

\[\begin{array}{l} \\ e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)} \end{array} \]
(FPCore (x s) :precision binary32 (exp (- (log1p (exp (/ (- x) s))))))
float code(float x, float s) {
	return expf(-log1pf(expf((-x / s))));
}

function code(x, s)
	return exp(Float32(-log1p(exp(Float32(Float32(-x) / s)))))
end

\begin{array}{l}

\\
e^{-\mathsf{log1p}\left(e^{\frac{-x}{s}}\right)}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \frac{1}{1 + {\left(e^{6}\right)}^{\left(-0.16666666666666666 \cdot \frac{x}{s}\right)}} \end{array} \]
(FPCore (x s)
 :precision binary32
 (/ 1.0 (+ 1.0 (pow (exp 6.0) (* -0.16666666666666666 (/ x s))))))
float code(float x, float s) {
	return 1.0f / (1.0f + powf(expf(6.0f), (-0.16666666666666666f * (x / s))));
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / (1.0e0 + (exp(6.0e0) ** ((-0.16666666666666666e0) * (x / s))))
end function

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(1.0) + (exp(Float32(6.0)) ^ Float32(Float32(-0.16666666666666666) * Float32(x / s)))))
end

function tmp = code(x, s)
	tmp = single(1.0) / (single(1.0) + (exp(single(6.0)) ^ (single(-0.16666666666666666) * (x / s))));
end

\begin{array}{l}

\\
\frac{1}{1 + {\left(e^{6}\right)}^{\left(-0.16666666666666666 \cdot \frac{x}{s}\right)}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup?

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

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

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

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

\begin{array}{l}

\\
\frac{1}{e^{\frac{-x}{s}} + 1}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 4: 49.0% accurate, 6.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.16666666666666666 \cdot \left(x \cdot \left(-6 \cdot \frac{1}{s}\right)\right)\right)}\\ \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34)
   0.5
   (/ 1.0 (+ 1.0 (+ 1.0 (* 0.16666666666666666 (* x (* -6.0 (/ 1.0 s)))))))))
float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.16666666666666666f * (x * (-6.0f * (1.0f / s))))));
	}
	return tmp;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-1.000000046701102e-34)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.16666666666666666e0 * (x * ((-6.0e0) * (1.0e0 / s))))))
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.000000046701102e-34))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) * Float32(x * Float32(Float32(-6.0) * Float32(Float32(1.0) / s)))))));
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.000000046701102e-34))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.16666666666666666) * (x * (single(-6.0) * (single(1.0) / s))))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\
\;\;\;\;0.5\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 + \left(1 + 0.16666666666666666 \cdot \left(x \cdot \left(-6 \cdot \frac{1}{s}\right)\right)\right)}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 49.0% accurate, 6.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 + \left(1 + 0.16666666666666666 \cdot \left(x \cdot \frac{-6}{s}\right)\right)}\\ \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34)
   0.5
   (/ 1.0 (+ 1.0 (+ 1.0 (* 0.16666666666666666 (* x (/ -6.0 s))))))))
float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (1.0f + (1.0f + (0.16666666666666666f * (x * (-6.0f / s)))));
	}
	return tmp;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-1.000000046701102e-34)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (1.0e0 + (1.0e0 + (0.16666666666666666e0 * (x * ((-6.0e0) / s)))))
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.000000046701102e-34))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(Float32(0.16666666666666666) * Float32(x * Float32(Float32(-6.0) / s))))));
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.000000046701102e-34))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(1.0) + (single(1.0) + (single(0.16666666666666666) * (x * (single(-6.0) / s)))));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\
\;\;\;\;0.5\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 + \left(1 + 0.16666666666666666 \cdot \left(x \cdot \frac{-6}{s}\right)\right)}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 48.4% accurate, 10.7× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 - \frac{x}{s}}\\ \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (if (<= (- x) -1.000000046701102e-34) 0.5 (/ 1.0 (- 2.0 (/ x s)))))
float code(float x, float s) {
	float tmp;
	if (-x <= -1.000000046701102e-34f) {
		tmp = 0.5f;
	} else {
		tmp = 1.0f / (2.0f - (x / s));
	}
	return tmp;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (-x <= (-1.000000046701102e-34)) then
        tmp = 0.5e0
    else
        tmp = 1.0e0 / (2.0e0 - (x / s))
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (Float32(-x) <= Float32(-1.000000046701102e-34))
		tmp = Float32(0.5);
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(2.0) - Float32(x / s)));
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (-x <= single(-1.000000046701102e-34))
		tmp = single(0.5);
	else
		tmp = single(1.0) / (single(2.0) - (x / s));
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;-x \leq -1.000000046701102 \cdot 10^{-34}:\\
\;\;\;\;0.5\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2 - \frac{x}{s}}\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 45.7% accurate, 17.6× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12}:\\ \;\;\;\;\frac{-s}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (if (<= x -3.999999984016789e-12) (/ (- s) x) 0.5))
float code(float x, float s) {
	float tmp;
	if (x <= -3.999999984016789e-12f) {
		tmp = -s / x;
	} else {
		tmp = 0.5f;
	}
	return tmp;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= (-3.999999984016789e-12)) then
        tmp = -s / x
    else
        tmp = 0.5e0
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(-3.999999984016789e-12))
		tmp = Float32(Float32(-s) / x);
	else
		tmp = Float32(0.5);
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(-3.999999984016789e-12))
		tmp = -s / x;
	else
		tmp = single(0.5);
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq -3.999999984016789 \cdot 10^{-12}:\\
\;\;\;\;\frac{-s}{x}\\

\mathbf{else}:\\
\;\;\;\;0.5\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 35.5% accurate, 108.0× speedup?

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

real(4) function code(x, s)
    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

\begin{array}{l}

\\
0.5
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023343 
(FPCore (x s)
  :name "Logistic function"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ 1.0 (+ 1.0 (exp (/ (- x) s)))))