Logistic distribution

Percentage Accurate: 99.6% → 99.6%
Time: 6.0s
Alternatives: 1
Speedup: 1.2×

Specification

?
\[0 \leq s \land s \leq 1.0651631\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))
float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := e^{\frac{-\left|x\right|}{s}}\\
t_1 := 1 + t\_0\\
\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1}
\end{array}
\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 1 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: 99.6% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{-\left|x\right|}{s}}\\ t_1 := 1 + t\_0\\ \frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1} \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (exp (/ (- (fabs x)) s))) (t_1 (+ 1.0 t_0)))
   (/ t_0 (* (* s t_1) t_1))))
float code(float x, float s) {
	float t_0 = expf((-fabsf(x) / s));
	float t_1 = 1.0f + t_0;
	return t_0 / ((s * t_1) * t_1);
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: t_0
    real(4) :: t_1
    t_0 = exp((-abs(x) / s))
    t_1 = 1.0e0 + t_0
    code = t_0 / ((s * t_1) * t_1)
end function

function code(x, s)
	t_0 = exp(Float32(Float32(-abs(x)) / s))
	t_1 = Float32(Float32(1.0) + t_0)
	return Float32(t_0 / Float32(Float32(s * t_1) * t_1))
end

function tmp = code(x, s)
	t_0 = exp((-abs(x) / s));
	t_1 = single(1.0) + t_0;
	tmp = t_0 / ((s * t_1) * t_1);
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := e^{\frac{-\left|x\right|}{s}}\\
t_1 := 1 + t\_0\\
\frac{t\_0}{\left(s \cdot t\_1\right) \cdot t\_1}
\end{array}
\end{array}

Alternative 1: 99.6% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left|x\right|}{s}\\ \frac{1}{\left(1 + e^{-t\_0}\right) \cdot \mathsf{fma}\left(s, e^{t\_0}, s\right)} \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (let* ((t_0 (/ (fabs x) s)))
   (/ 1.0 (* (+ 1.0 (exp (- t_0))) (fma s (exp t_0) s)))))
float code(float x, float s) {
	float t_0 = fabsf(x) / s;
	return 1.0f / ((1.0f + expf(-t_0)) * fmaf(s, expf(t_0), s));
}

function code(x, s)
	t_0 = Float32(abs(x) / s)
	return Float32(Float32(1.0) / Float32(Float32(Float32(1.0) + exp(Float32(-t_0))) * fma(s, exp(t_0), s)))
end

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{\left|x\right|}{s}\\
\frac{1}{\left(1 + e^{-t\_0}\right) \cdot \mathsf{fma}\left(s, e^{t\_0}, s\right)}
\end{array}
\end{array}
Derivation

  1. Initial program 99.5%

    \[\frac{e^{\frac{-\left|x\right|}{s}}}{\left(s \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)\right) \cdot \left(1 + e^{\frac{-\left|x\right|}{s}}\right)} \]

  3. Simplified99.5%

    \[\leadsto \color{blue}{\frac{1}{\left(1 + e^{\frac{-\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)}} \]
  4. Add Preprocessing

  5. Final simplification99.5%

    \[\leadsto \frac{1}{\left(1 + e^{-\frac{\left|x\right|}{s}}\right) \cdot \mathsf{fma}\left(s, e^{\frac{\left|x\right|}{s}}, s\right)} \]
  6. Add Preprocessing

Reproduce

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herbie shell --seed 2024039 
(FPCore (x s)
  :name "Logistic distribution"
  :precision binary32
  :pre (and (<= 0.0 s) (<= s 1.0651631))
  (/ (exp (/ (- (fabs x)) s)) (* (* s (+ 1.0 (exp (/ (- (fabs x)) s)))) (+ 1.0 (exp (/ (- (fabs x)) s))))))