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

Average Accuracy: 60.8% → 98.4%

Time: 18.1s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z t) :precision binary64 (- x (/ (log1p (* y (expm1 z))) 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) {
	return x - (log1p((y * expm1(z))) / t);
}

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) {
	return x - (Math.log1p((y * Math.expm1(z))) / t);
}

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):
	return x - (math.log1p((y * math.expm1(z))) / t)

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)
	return Float64(x - Float64(log1p(Float64(y * expm1(z))) / t))
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_] := N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Target

Original	60.8%
Target	74.3%
Herbie	98.4%

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

Derivation

1. Initial program 60.8%

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

2. Simplified 98.4%

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

   [Start] 60.8	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   *-lft-identity [<=] 60.8	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
   distribute-lft-out-- [<=] 60.8	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
   *-lft-identity [=>] 60.8	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   *-commutative [<=] 60.8	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
   *-rgt-identity [=>] 60.8	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]

3. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	93.8%
Cost	13764

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 10^{-12}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]

Alternative 2
Accuracy	92.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -150000 \lor \neg \left(y \leq 50000000\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \end{array} \]

Alternative 3
Accuracy	84.6%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+103}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.7 \cdot 10^{+169}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]

Alternative 4
Accuracy	81.8%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{y}{\frac{t}{z \cdot z}} \cdot -0.5 - \frac{y}{\frac{t}{z}}\right)\\ \end{array} \]

Alternative 5
Accuracy	78.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -6800:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 6
Accuracy	82.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 7
Accuracy	71.4%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023141 
(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)))