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

Average Error: 25.1 → 0.4

Time: 18.1s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-72} \lor \neg \left(y \leq 10^{-23}\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1e-72) (not (<= y 1e-23)))
   (- x (/ (log1p (* y (expm1 z))) t))
   (- x (/ (expm1 z) (/ t y)))))

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 tmp;
	if ((y <= -1e-72) || !(y <= 1e-23)) {
		tmp = x - (log1p((y * expm1(z))) / t);
	} else {
		tmp = x - (expm1(z) / (t / y));
	}
	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 tmp;
	if ((y <= -1e-72) || !(y <= 1e-23)) {
		tmp = x - (Math.log1p((y * Math.expm1(z))) / t);
	} else {
		tmp = x - (Math.expm1(z) / (t / y));
	}
	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):
	tmp = 0
	if (y <= -1e-72) or not (y <= 1e-23):
		tmp = x - (math.log1p((y * math.expm1(z))) / t)
	else:
		tmp = x - (math.expm1(z) / (t / y))
	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)
	tmp = 0.0
	if ((y <= -1e-72) || !(y <= 1e-23))
		tmp = Float64(x - Float64(log1p(Float64(y * expm1(z))) / t));
	else
		tmp = Float64(x - Float64(expm1(z) / Float64(t / y)));
	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_] := If[Or[LessEqual[y, -1e-72], N[Not[LessEqual[y, 1e-23]], $MachinePrecision]], N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	25.1
Target	15.9
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if y < -9.9999999999999997e-73 or 9.9999999999999996e-24 < y

   1. Initial program 39.4

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

   2. Simplified 0.6

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

      [Start] 39.4	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-lft-identity [<=] 39.4	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
      distribute-lft-out-- [<=] 39.4	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
      *-lft-identity [=>] 39.4	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-commutative [<=] 39.4	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
      *-rgt-identity [=>] 39.4	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]

   if -9.9999999999999997e-73 < y < 9.9999999999999996e-24

   1. Initial program 10.6

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

   2. Simplified 1.4

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

      [Start] 10.6	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-lft-identity [<=] 10.6	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
      distribute-lft-out-- [<=] 10.6	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
      *-lft-identity [=>] 10.6	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-commutative [<=] 10.6	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
      *-rgt-identity [=>] 10.6	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]

   3. Taylor expanded in y around 0 4.9

      \[\leadsto x - \color{blue}{\frac{\left(e^{z} - 1\right) \cdot y}{t}} \]

   4. Simplified 0.2

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

      [Start] 4.9	\[ x - \frac{\left(e^{z} - 1\right) \cdot y}{t} \]
      associate-/l* [=>] 4.9	\[ x - \color{blue}{\frac{e^{z} - 1}{\frac{t}{y}}} \]
      expm1-def [=>] 0.2	\[ x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{\frac{t}{y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-72} \lor \neg \left(y \leq 10^{-23}\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.5
Cost	7497

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-15} \lor \neg \left(y \leq 10^{-284}\right):\\ \;\;\;\;x - \frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \end{array} \]

Alternative 2
Error	9.4
Cost	6848

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

Alternative 3
Error	18.4
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-205}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-252}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	11.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -6400:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 5
Error	11.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -3500:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 6
Error	18.0
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023066 
(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)))