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

Average Error: 24.8 → 1.2

Time: 16.5s

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	24.8
Target	16.4
Herbie	1.2

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

Derivation

1. Initial program 24.8

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

2. Simplified 1.2

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

   [Start] 24.8	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   *-lft-identity [<=] 24.8	\[ \color{blue}{1 \cdot \left(x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)} \]
   distribute-lft-out-- [<=] 24.8	\[ \color{blue}{1 \cdot x - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
   *-lft-identity [=>] 24.8	\[ \color{blue}{x} - 1 \cdot \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   *-commutative [<=] 24.8	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \cdot 1} \]
   *-rgt-identity [=>] 24.8	\[ x - \color{blue}{\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}} \]
   associate-+l- [=>] 14.7	\[ x - \frac{\log \color{blue}{\left(1 - \left(y - y \cdot e^{z}\right)\right)}}{t} \]
   sub-neg [=>] 14.7	\[ x - \frac{\log \color{blue}{\left(1 + \left(-\left(y - y \cdot e^{z}\right)\right)\right)}}{t} \]
   log1p-def [=>] 10.9	\[ x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]
   sub-neg [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(-\color{blue}{\left(y + \left(-y \cdot e^{z}\right)\right)}\right)}{t} \]
   distribute-neg-in [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right) + \left(-\left(-y \cdot e^{z}\right)\right)}\right)}{t} \]
   +-commutative [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-\left(-y \cdot e^{z}\right)\right) + \left(-y\right)}\right)}{t} \]
   remove-double-neg [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z}} + \left(-y\right)\right)}{t} \]
   *-commutative [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{e^{z} \cdot y} + \left(-y\right)\right)}{t} \]
   neg-mul-1 [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(e^{z} \cdot y + \color{blue}{-1 \cdot y}\right)}{t} \]
   distribute-rgt-out [=>] 10.9	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} + -1\right)}\right)}{t} \]
   metadata-eval [<=] 10.9	\[ x - \frac{\mathsf{log1p}\left(y \cdot \left(e^{z} + \color{blue}{\left(-1\right)}\right)\right)}{t} \]
   sub-neg [<=] 10.9	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]
   expm1-def [=>] 1.2	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

3. Final simplification 1.2

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

Alternatives

Alternative 1
Error	4.5
Cost	7364

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]

Alternative 2
Error	5.3
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1600 \lor \neg \left(y \leq 0.0058\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 3
Error	9.0
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{+134}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+48}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	19.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-179}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-287}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	18.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -1.46 \cdot 10^{-229}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-251}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	11.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 7
Error	18.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022364 
(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)))