System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Average Error: 24.6 → 1.4

Time: 13.1s

Precision: binary64

Math

\[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

↓

\[\begin{array}{l} t_1 := x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{if}\;z \leq -6.255462246592998 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.0409400238575313 \cdot 10^{-194}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log1p (* y (expm1 z))) t))))
   (if (<= z -6.255462246592998e-193)
     t_1
     (if (<= z 1.0409400238575313e-194) (- x (/ y (/ t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / t);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = x - (log1p((y * expm1(z))) / t);
	double tmp;
	if (z <= -6.255462246592998e-193) {
		tmp = t_1;
	} else if (z <= 1.0409400238575313e-194) {
		tmp = x - (y / (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log(((1.0 - y) + (y * Math.exp(z)))) / t);
}

↓

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (Math.log1p((y * Math.expm1(z))) / t);
	double tmp;
	if (z <= -6.255462246592998e-193) {
		tmp = t_1;
	} else if (z <= 1.0409400238575313e-194) {
		tmp = x - (y / (t / z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	return x - (math.log(((1.0 - y) + (y * math.exp(z)))) / t)

↓

def code(x, y, z, t):
	t_1 = x - (math.log1p((y * math.expm1(z))) / t)
	tmp = 0
	if z <= -6.255462246592998e-193:
		tmp = t_1
	elif z <= 1.0409400238575313e-194:
		tmp = x - (y / (t / z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log1p(Float64(y * expm1(z))) / t))
	tmp = 0.0
	if (z <= -6.255462246592998e-193)
		tmp = t_1;
	elseif (z <= 1.0409400238575313e-194)
		tmp = Float64(x - Float64(y / Float64(t / z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.255462246592998e-193], t$95$1, If[LessEqual[z, 1.0409400238575313e-194], N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	24.6
Target	15.8
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{\frac{-0.5}{y \cdot t}}{z \cdot z}\right) - \frac{-0.5}{y \cdot t} \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if z < -6.255462246592998e-193 or 1.0409400238575313e-194 < z

   1. Initial program 22.4

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Simplified 0.9

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]

   if -6.255462246592998e-193 < z < 1.0409400238575313e-194

   1. Initial program 30.4

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]

   2. Simplified 1.1

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]

   3. Applied egg-rr 6.0

      \[\leadsto x - \frac{\color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)\right)}^{3}}}}{t} \]

   4. Taylor expanded in z around 0 3.1

      \[\leadsto x - \color{blue}{\frac{y \cdot z}{t}} \]

   5. Simplified 2.6

      \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.255462246592998 \cdot 10^{-193}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{elif}\;z \leq 1.0409400238575313 \cdot 10^{-194}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))