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

Average Error: 24.6 → 1.0

Time: 19.2s

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	24.6
Target	16.4
Herbie	1.0

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

Derivation

1. Initial program 24.6

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

2. Simplified 1.0

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

3. Final simplification 1.0

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

Alternatives

Alternative 1
Error	4.5
Cost	13448

\[\begin{array}{l} \mathbf{if}\;y \leq -173.48997110354924:\\ \;\;\;\;x - \frac{1}{t \cdot 0.5 + \frac{t}{\left(1 + z \cdot 0.5\right) \cdot \left(y \cdot z\right)}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{expm1}\left(z\right), \frac{-y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot 0.5\right)\right)\right)}{t}\\ \end{array} \]

Alternative 2
Error	4.5
Cost	7496

\[\begin{array}{l} \mathbf{if}\;y \leq -173.48997110354924:\\ \;\;\;\;x - \frac{1}{t \cdot 0.5 + \frac{t}{\left(1 + z \cdot 0.5\right) \cdot \left(y \cdot z\right)}}\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{+24}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot 0.5\right)\right)\right)}{t}\\ \end{array} \]

Alternative 3
Error	7.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;y \leq -173.48997110354924:\\ \;\;\;\;x - \frac{1}{t \cdot 0.5 + \frac{t}{\left(1 + z \cdot 0.5\right) \cdot \left(y \cdot z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \end{array} \]

Alternative 4
Error	11.3
Cost	1480

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.680952625312589 \cdot 10^{-233}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{t \cdot 0.5 + \frac{t}{\left(1 + z \cdot 0.5\right) \cdot \left(y \cdot z\right)}}\\ \end{array} \]

Alternative 5
Error	19.0
Cost	648

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3865486965715162 \cdot 10^{-282}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8516088628711375 \cdot 10^{-173}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	11.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 7
Error	18.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022228 
(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)))