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

Average Error: 25.0 → 1.1

Time: 1.8min

Precision: binary64

Cost: 13248

Math

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

↓

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

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 (* (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) {
	return x - (log1p((expm1(z) * y)) / 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((Math.expm1(z) * y)) / 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((math.expm1(z) * y)) / 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(expm1(z) * y)) / 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[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Target

Original	25.0
Target	16.2
Herbie	1.1

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

Derivation

1. Initial program 25.0

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

2. Simplified 11.3

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

3. Applied egg-rr 1.1

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

Alternatives

Alternative 1
Error	1.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;z \leq -3700000000:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(-y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot y\right)}{t}\\ \end{array} \]

Alternative 2
Error	4.7
Cost	6916

\[\begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-11}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(-y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \begin{array}{l} \mathbf{if}\;z \ne 0:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \end{array}\\ \end{array} \]

Alternative 3
Error	20.4
Cost	1240

\[\begin{array}{l} t_1 := \frac{-y \cdot z}{t}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.4 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{-296}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-208}:\\ \;\;\;\;\frac{-1}{t} \cdot \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	20.4
Cost	1176

\[\begin{array}{l} t_1 := \frac{-y \cdot z}{t}\\ \mathbf{if}\;x \leq -1.75 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-153}:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-296}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	9.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x - \begin{array}{l} \mathbf{if}\;z \ne 0:\\ \;\;\;\;\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{t}\\ \end{array}\\ \end{array} \]

Alternative 6
Error	11.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{t} \cdot z\\ \end{array} \]

Alternative 7
Error	9.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -850000:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \]

Alternative 8
Error	16.3
Cost	452

\[\begin{array}{l} \mathbf{if}\;z \leq -85000000000000:\\ \;\;\;\;\frac{y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	18.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023033 
(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)))