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

Average Error: 24.9 → 1.0

Time: 16.2s

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	24.9
Target	16.6
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 24.9

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

2. Simplified 1.0

   \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   Proof
   (-.f64 x (/.f64 (log1p.f64 (*.f64 y (expm1.f64 z))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (*.f64 y (Rewrite<= expm1-def_binary64 (-.f64 (exp.f64 z) 1)))) t)): 47 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (*.f64 y (Rewrite=> sub-neg_binary64 (+.f64 (exp.f64 z) (neg.f64 1))))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (*.f64 y (+.f64 (exp.f64 z) (Rewrite=> metadata-eval -1)))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 -1 (exp.f64 z))))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 y -1) (*.f64 y (exp.f64 z))))) t)): 2 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 y)) (*.f64 y (exp.f64 z)))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 y)) (*.f64 y (exp.f64 z)))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 y)) (*.f64 y (exp.f64 z)))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 y (*.f64 y (exp.f64 z)))))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log1p.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 y (*.f64 y (exp.f64 z)))))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (neg.f64 (-.f64 y (*.f64 y (exp.f64 z))))))) t)): 14 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (-.f64 y (*.f64 y (exp.f64 z)))))) t)): 0 points increase in error, 0 points decrease in error
   (-.f64 x (/.f64 (log.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 1 y) (*.f64 y (exp.f64 z))))) t)): 69 points increase in error, 0 points decrease in error

3. Final simplification 1.0

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

Alternatives

Alternative 1
Error	3.8
Cost	13764

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(y \cdot 0.5\right)\right)\right)}{t}\\ \end{array} \]

Alternative 2
Error	6.5
Cost	7232

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

Alternative 3
Error	8.7
Cost	6848

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

Alternative 4
Error	11.1
Cost	1220

\[\begin{array}{l} \mathbf{if}\;z \leq -7400000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{y}{\frac{t}{z}} + 0.5 \cdot \frac{y}{\frac{t}{z \cdot z}}\right)\\ \end{array} \]

Alternative 5
Error	18.2
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-211}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-257}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	13.5
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7
Error	11.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+14}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 8
Error	18.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022330 
(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)))