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

Average Error: 39.45% → 0.49%

Time: 16.2s

Precision: binary64

Cost: 13513

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8} \lor \neg \left(y \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \end{array} \]

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 -7.2e-8) (not (<= y 5e-77)))
   (- x (/ (log1p (* y (expm1 z))) t))
   (- x (* (expm1 z) (/ y t)))))

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 <= -7.2e-8) || !(y <= 5e-77)) {
		tmp = x - (log1p((y * expm1(z))) / t);
	} else {
		tmp = x - (expm1(z) * (y / t));
	}
	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 <= -7.2e-8) || !(y <= 5e-77)) {
		tmp = x - (Math.log1p((y * Math.expm1(z))) / t);
	} else {
		tmp = x - (Math.expm1(z) * (y / t));
	}
	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 <= -7.2e-8) or not (y <= 5e-77):
		tmp = x - (math.log1p((y * math.expm1(z))) / t)
	else:
		tmp = x - (math.expm1(z) * (y / t))
	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 <= -7.2e-8) || !(y <= 5e-77))
		tmp = Float64(x - Float64(log1p(Float64(y * expm1(z))) / t));
	else
		tmp = Float64(x - Float64(expm1(z) * Float64(y / t)));
	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, -7.2e-8], N[Not[LessEqual[y, 5e-77]], $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[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	39.45%
Target	25.39%
Herbie	0.49%

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

Derivation

1. Split input into 2 regimes

2. if y < -7.19999999999999962e-8 or 4.99999999999999963e-77 < y

   1. Initial program 62.68

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

   2. Simplified 0.72

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

      [Start] 62.68	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-lft-identity [<=] 62.68	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
      distribute-lft-out-- [<=] 62.68	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
      *-lft-identity [=>] 62.68	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-commutative [<=] 62.68	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
      *-rgt-identity [=>] 62.68	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]

   if -7.19999999999999962e-8 < y < 4.99999999999999963e-77

   1. Initial program 17.33

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

   2. Simplified 2.11

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

      [Start] 17.33	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-lft-identity [<=] 17.33	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
      distribute-lft-out-- [<=] 17.33	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
      *-lft-identity [=>] 17.33	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      *-commutative [<=] 17.33	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
      *-rgt-identity [=>] 17.33	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]

   3. Taylor expanded in y around 0 7.86

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

   4. Simplified 0.31

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

      [Start] 7.86	\[ x - \frac{\left(e^{z} - 1\right) \cdot y}{t} \]
      associate-/l* [=>] 7.9	\[ x - \color{blue}{\frac{e^{z} - 1}{\frac{t}{y}}} \]
      expm1-def [=>] 0.31	\[ x - \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{\frac{t}{y}} \]

   5. Taylor expanded in z around inf 7.86

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

   6. Simplified 0.27

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

      [Start] 7.86	\[ x - \frac{\left(e^{z} - 1\right) \cdot y}{t} \]
      *-commutative [=>] 7.86	\[ x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
      expm1-def [=>] 2.17	\[ x - \frac{y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]
      associate-*l/ [<=] 0.27	\[ x - \color{blue}{\frac{y}{t} \cdot \mathsf{expm1}\left(z\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.49

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-8} \lor \neg \left(y \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.95%
Cost	7232

\[x - \frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5} \]

Alternative 2
Error	13.73%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+155}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 3
Error	19.1%
Cost	1088

\[x - \frac{y}{0.25 \cdot \left(z \cdot t\right) + \left(\frac{t}{z} + t \cdot -0.5\right)} \]

Alternative 4
Error	19.12%
Cost	960

\[x - \frac{y}{t \cdot \left(-0.5 + \left(z \cdot 0.25 + \frac{1}{z}\right)\right)} \]

Alternative 5
Error	19.22%
Cost	832

\[x - \frac{y}{0.25 \cdot \left(z \cdot t\right) + \frac{t}{z}} \]

Alternative 6
Error	22.18%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -0.0024:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7
Error	18.86%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -310:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 8
Error	29.04%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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)))