Logistic distribution

Percentage Accurate: 99.5% → 99.6%
Time: 11.5s
Alternatives: 10
Speedup: 1.0×

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 10 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: 99.5% 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, 1.0× speedup?

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

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

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

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

\begin{array}{l}

\\
\begin{array}{l}
t_0 := e^{\frac{-\left|x\right|}{s}}\\
\frac{t_0}{\left(t_0 + 1\right) \cdot \left(s + \frac{s}{e^{\frac{\left|x\right|}{s}}}\right)}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 2: 49.3% accurate, 1.0× speedup?

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

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

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

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

\begin{array}{l}

\\
\begin{array}{l}
t_0 := \sqrt{\frac{x}{s}}\\
\frac{1}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(1 + {\left(e^{t_0}\right)}^{t_0}\right)\right)}
\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 3: 61.9% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\left(e^{\frac{-\left|x\right|}{s}} + 1\right) \cdot \left(1 + e^{\frac{x}{s}}\right)\right)} \end{array} \]
(FPCore (x s)
 :precision binary32
 (/ 1.0 (* s (* (+ (exp (/ (- (fabs x)) s)) 1.0) (+ 1.0 (exp (/ x s)))))))
float code(float x, float s) {
	return 1.0f / (s * ((expf((-fabsf(x) / s)) + 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 / (s * ((exp((-abs(x) / s)) + 1.0e0) * (1.0e0 + exp((x / s)))))
end function

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

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

\begin{array}{l}

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

Alternative 4: 95.7% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \frac{e^{\frac{-\left|x\right|}{s}}}{\left|x\right| \cdot -4 + s \cdot 4} \end{array} \]
(FPCore (x s)
 :precision binary32
 (/ (exp (/ (- (fabs x)) s)) (+ (* (fabs x) -4.0) (* s 4.0))))
float code(float x, float s) {
	return expf((-fabsf(x) / s)) / ((fabsf(x) * -4.0f) + (s * 4.0f));
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = exp((-abs(x) / s)) / ((abs(x) * (-4.0e0)) + (s * 4.0e0))
end function

function code(x, s)
	return Float32(exp(Float32(Float32(-abs(x)) / s)) / Float32(Float32(abs(x) * Float32(-4.0)) + Float32(s * Float32(4.0))))
end

function tmp = code(x, s)
	tmp = exp((-abs(x) / s)) / ((abs(x) * single(-4.0)) + (s * single(4.0)));
end

\begin{array}{l}

\\
\frac{e^{\frac{-\left|x\right|}{s}}}{\left|x\right| \cdot -4 + s \cdot 4}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 5: 59.4% accurate, 3.0× speedup?

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{{\left(1 + e^{\frac{x}{s}}\right)}^{2}} \end{array} \]
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (pow (+ 1.0 (exp (/ x s))) 2.0)))
float code(float x, float s) {
	return (1.0f / s) / powf((1.0f + expf((x / s))), 2.0f);
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / ((1.0e0 + exp((x / s))) ** 2.0e0)
end function

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / (Float32(Float32(1.0) + exp(Float32(x / s))) ^ Float32(2.0)))
end

function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((single(1.0) + exp((x / s))) ^ single(2.0));
end

\begin{array}{l}

\\
\frac{\frac{1}{s}}{{\left(1 + e^{\frac{x}{s}}\right)}^{2}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 6: 73.3% accurate, 5.5× speedup?

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{{\left(1 + \left(1 + \frac{x}{s}\right)\right)}^{2}} \end{array} \]
(FPCore (x s)
 :precision binary32
 (/ (/ 1.0 s) (pow (+ 1.0 (+ 1.0 (/ x s))) 2.0)))
float code(float x, float s) {
	return (1.0f / s) / powf((1.0f + (1.0f + (x / s))), 2.0f);
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / ((1.0e0 + (1.0e0 + (x / s))) ** 2.0e0)
end function

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / (Float32(Float32(1.0) + Float32(Float32(1.0) + Float32(x / s))) ^ Float32(2.0)))
end

function tmp = code(x, s)
	tmp = (single(1.0) / s) / ((single(1.0) + (single(1.0) + (x / s))) ^ single(2.0));
end

\begin{array}{l}

\\
\frac{\frac{1}{s}}{{\left(1 + \left(1 + \frac{x}{s}\right)\right)}^{2}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 7: 50.4% accurate, 56.4× speedup?

\[\begin{array}{l} \\ \frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot 4} \end{array} \]
(FPCore (x s) :precision binary32 (/ (/ 1.0 s) (+ 4.0 (* (/ x s) 4.0))))
float code(float x, float s) {
	return (1.0f / s) / (4.0f + ((x / s) * 4.0f));
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = (1.0e0 / s) / (4.0e0 + ((x / s) * 4.0e0))
end function

function code(x, s)
	return Float32(Float32(Float32(1.0) / s) / Float32(Float32(4.0) + Float32(Float32(x / s) * Float32(4.0))))
end

function tmp = code(x, s)
	tmp = (single(1.0) / s) / (single(4.0) + ((x / s) * single(4.0)));
end

\begin{array}{l}

\\
\frac{\frac{1}{s}}{4 + \frac{x}{s} \cdot 4}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 8: 64.7% accurate, 56.4× speedup?

\[\begin{array}{l} \\ \frac{1}{s \cdot 4 + \frac{x \cdot x}{s}} \end{array} \]
(FPCore (x s) :precision binary32 (/ 1.0 (+ (* s 4.0) (/ (* x x) s))))
float code(float x, float s) {
	return 1.0f / ((s * 4.0f) + ((x * x) / s));
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 1.0e0 / ((s * 4.0e0) + ((x * x) / s))
end function

function code(x, s)
	return Float32(Float32(1.0) / Float32(Float32(s * Float32(4.0)) + Float32(Float32(x * x) / s)))
end

function tmp = code(x, s)
	tmp = single(1.0) / ((s * single(4.0)) + ((x * x) / s));
end

\begin{array}{l}

\\
\frac{1}{s \cdot 4 + \frac{x \cdot x}{s}}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 28.8% accurate, 121.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0007999999797903001:\\ \;\;\;\;\frac{0.25}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{x}\\ \end{array} \end{array} \]
(FPCore (x s)
 :precision binary32
 (if (<= x 0.0007999999797903001) (/ 0.25 s) (/ 0.3333333333333333 x)))
float code(float x, float s) {
	float tmp;
	if (x <= 0.0007999999797903001f) {
		tmp = 0.25f / s;
	} else {
		tmp = 0.3333333333333333f / x;
	}
	return tmp;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    real(4) :: tmp
    if (x <= 0.0007999999797903001e0) then
        tmp = 0.25e0 / s
    else
        tmp = 0.3333333333333333e0 / x
    end if
    code = tmp
end function

function code(x, s)
	tmp = Float32(0.0)
	if (x <= Float32(0.0007999999797903001))
		tmp = Float32(Float32(0.25) / s);
	else
		tmp = Float32(Float32(0.3333333333333333) / x);
	end
	return tmp
end

function tmp_2 = code(x, s)
	tmp = single(0.0);
	if (x <= single(0.0007999999797903001))
		tmp = single(0.25) / s;
	else
		tmp = single(0.3333333333333333) / x;
	end
	tmp_2 = tmp;
end

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;x \leq 0.0007999999797903001:\\
\;\;\;\;\frac{0.25}{s}\\

\mathbf{else}:\\
\;\;\;\;\frac{0.3333333333333333}{x}\\


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

Alternative 10: 27.2% accurate, 206.7× speedup?

\[\begin{array}{l} \\ \frac{0.25}{s} \end{array} \]
(FPCore (x s) :precision binary32 (/ 0.25 s))
float code(float x, float s) {
	return 0.25f / s;
}

real(4) function code(x, s)
    real(4), intent (in) :: x
    real(4), intent (in) :: s
    code = 0.25e0 / s
end function

function code(x, s)
	return Float32(Float32(0.25) / s)
end

function tmp = code(x, s)
	tmp = single(0.25) / s;
end

\begin{array}{l}

\\
\frac{0.25}{s}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2023350 
(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))))))