Sample trimmed logistic on [-pi, pi]

Average Error: 0.3 → 0.3

Time: 40.3s

Precision: binary64

Cost: 162176

\[\left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 1\right) \land \left(0 \leq s \land s \leq 1.0651631\right)\]

Math

\[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

↓

\[\begin{array}{l} t_0 := e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\\ t_1 := \left(t_0 - e^{-\mathsf{log1p}\left(e^{\frac{\pi}{-s}}\right)}\right) \cdot u - t_0\\ \left(-s\right) \cdot \log \left(\frac{-1 - t_1}{t_1}\right) \end{array} \]

FPCore

(FPCore (u s)
 :precision binary64
 (*
  (- s)
  (log
   (-
    (/
     1.0
     (+
      (*
       u
       (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s))))))
      (/ 1.0 (+ 1.0 (exp (/ PI s))))))
    1.0))))

↓

(FPCore (u s)
 :precision binary64
 (let* ((t_0 (exp (- (log1p (exp (/ PI s))))))
        (t_1 (- (* (- t_0 (exp (- (log1p (exp (/ PI (- s))))))) u) t_0)))
   (* (- s) (log (/ (- -1.0 t_1) t_1)))))

C

double code(double u, double s) {
	return -s * log(((1.0 / ((u * ((1.0 / (1.0 + exp((-((double) M_PI) / s)))) - (1.0 / (1.0 + exp((((double) M_PI) / s)))))) + (1.0 / (1.0 + exp((((double) M_PI) / s)))))) - 1.0));
}

↓

double code(double u, double s) {
	double t_0 = exp(-log1p(exp((((double) M_PI) / s))));
	double t_1 = ((t_0 - exp(-log1p(exp((((double) M_PI) / -s))))) * u) - t_0;
	return -s * log(((-1.0 - t_1) / t_1));
}

Java

public static double code(double u, double s) {
	return -s * Math.log(((1.0 / ((u * ((1.0 / (1.0 + Math.exp((-Math.PI / s)))) - (1.0 / (1.0 + Math.exp((Math.PI / s)))))) + (1.0 / (1.0 + Math.exp((Math.PI / s)))))) - 1.0));
}

↓

public static double code(double u, double s) {
	double t_0 = Math.exp(-Math.log1p(Math.exp((Math.PI / s))));
	double t_1 = ((t_0 - Math.exp(-Math.log1p(Math.exp((Math.PI / -s))))) * u) - t_0;
	return -s * Math.log(((-1.0 - t_1) / t_1));
}

Python

def code(u, s):
	return -s * math.log(((1.0 / ((u * ((1.0 / (1.0 + math.exp((-math.pi / s)))) - (1.0 / (1.0 + math.exp((math.pi / s)))))) + (1.0 / (1.0 + math.exp((math.pi / s)))))) - 1.0))

↓

def code(u, s):
	t_0 = math.exp(-math.log1p(math.exp((math.pi / s))))
	t_1 = ((t_0 - math.exp(-math.log1p(math.exp((math.pi / -s))))) * u) - t_0
	return -s * math.log(((-1.0 - t_1) / t_1))

Julia

function code(u, s)
	return Float64(Float64(-s) * log(Float64(Float64(1.0 / Float64(Float64(u * Float64(Float64(1.0 / Float64(1.0 + exp(Float64(Float64(-pi) / s)))) - Float64(1.0 / Float64(1.0 + exp(Float64(pi / s)))))) + Float64(1.0 / Float64(1.0 + exp(Float64(pi / s)))))) - 1.0)))
end

↓

function code(u, s)
	t_0 = exp(Float64(-log1p(exp(Float64(pi / s)))))
	t_1 = Float64(Float64(Float64(t_0 - exp(Float64(-log1p(exp(Float64(pi / Float64(-s))))))) * u) - t_0)
	return Float64(Float64(-s) * log(Float64(Float64(-1.0 - t_1) / t_1)))
end

Wolfram

code[u_, s_] := N[((-s) * N[Log[N[(N[(1.0 / N[(N[(u * N[(N[(1.0 / N[(1.0 + N[Exp[N[((-Pi) / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + N[Exp[N[(Pi / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(1.0 + N[Exp[N[(Pi / s), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[u_, s_] := Block[{t$95$0 = N[Exp[(-N[Log[1 + N[Exp[N[(Pi / s), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 - N[Exp[(-N[Log[1 + N[Exp[N[(Pi / (-s)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] * u), $MachinePrecision] - t$95$0), $MachinePrecision]}, N[((-s) * N[Log[N[(N[(-1.0 - t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 0.3

   \[\left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - \frac{1}{1 + e^{\frac{\pi}{s}}}\right) + \frac{1}{1 + e^{\frac{\pi}{s}}}} - 1\right) \]

2. Applied egg-rr 0.3

   \[\leadsto \left(-s\right) \cdot \log \color{blue}{\left(\frac{-1 - \left(\left(e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\pi}{-s}}\right)}\right) \cdot u - e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}\right)}{\left(e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)} - e^{-\mathsf{log1p}\left(e^{\frac{\pi}{-s}}\right)}\right) \cdot u - e^{-\mathsf{log1p}\left(e^{\frac{\pi}{s}}\right)}}\right)} \]

Alternatives

Alternative 1
Error	0.3
Cost	46720

\[\begin{array}{l} t_0 := \frac{1}{1 + e^{\frac{\pi}{s}}}\\ \left(-s\right) \cdot \log \left(\frac{1}{u \cdot \left(\frac{1}{1 + e^{\frac{-\pi}{s}}} - t_0\right) + t_0} - 1\right) \end{array} \]

Alternative 2
Error	54.4
Cost	13828

\[\begin{array}{l} t_0 := \mathsf{log1p}\left(\frac{1}{s}\right)\\ t_1 := t_0 \cdot \left(-s\right)\\ \mathbf{if}\;t_1 \ne 0:\\ \;\;\;\;\frac{2}{\frac{\frac{-2}{s}}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	54.4
Cost	6912

\[\left(-s \cdot s\right) + s \cdot \log s \]

Alternative 4
Error	54.4
Cost	6784

\[\left(-\mathsf{log1p}\left(\frac{1}{s}\right)\right) \cdot s \]

Alternative 5
Error	54.4
Cost	6592

\[s \cdot \log s \]

Alternative 6
Error	60.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (u s)
  :name "Sample trimmed logistic on [-pi, pi]"
  :precision binary64
  :pre (and (and (<= 2.328306437e-10 u) (<= u 1.0)) (and (<= 0.0 s) (<= s 1.0651631)))
  (* (- s) (log (- (/ 1.0 (+ (* u (- (/ 1.0 (+ 1.0 (exp (/ (- PI) s)))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) (/ 1.0 (+ 1.0 (exp (/ PI s)))))) 1.0))))