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

Percentage Accurate: 52.7% → 91.6%

Time: 45.1s

Precision: binary64

Cost: 20036

• could not determine a ground truth

  1. x = 2.387501152545511e-262
  2. y = 1.2178575715387347e-86
  3. z = -1.3729119915258474e+173
  4. t = -5.053792087175491e+68

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 1.0000006:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log z + \left(z \cdot 0.5 + \log y\right)}{t}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 1.0000006)
   (- x (/ (log1p (* y (expm1 z))) t))
   (- x (/ (+ (log z) (+ (* z 0.5) (log 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) {
	double tmp;
	if (exp(z) <= 1.0000006) {
		tmp = x - (log1p((y * expm1(z))) / t);
	} else {
		tmp = x - ((log(z) + ((z * 0.5) + log(y))) / t);
	}
	return tmp;
}

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) {
	double tmp;
	if (Math.exp(z) <= 1.0000006) {
		tmp = x - (Math.log1p((y * Math.expm1(z))) / t);
	} else {
		tmp = x - ((Math.log(z) + ((z * 0.5) + Math.log(y))) / t);
	}
	return tmp;
}

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):
	tmp = 0
	if math.exp(z) <= 1.0000006:
		tmp = x - (math.log1p((y * math.expm1(z))) / t)
	else:
		tmp = x - ((math.log(z) + ((z * 0.5) + math.log(y))) / t)
	return tmp

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)
	tmp = 0.0
	if (exp(z) <= 1.0000006)
		tmp = Float64(x - Float64(log1p(Float64(y * expm1(z))) / t));
	else
		tmp = Float64(x - Float64(Float64(log(z) + Float64(Float64(z * 0.5) + log(y))) / t));
	end
	return tmp
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_] := If[LessEqual[N[Exp[z], $MachinePrecision], 1.0000006], N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[Log[z], $MachinePrecision] + N[(N[(z * 0.5), $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Target

Original	52.7%
Target	53.1%
Herbie	91.6%

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 1.0000006

   1. Initial program 60.7%

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

   2. Simplified 96.4%

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

      [Start] 60.7%	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      associate-+l- [=>] 67.9%	\[ x - \frac{\log \color{blue}{\left(1 - \left(y - y \cdot e^{z}\right)\right)}}{t} \]
      sub-neg [=>] 67.9%	\[ x - \frac{\log \color{blue}{\left(1 + \left(-\left(y - y \cdot e^{z}\right)\right)\right)}}{t} \]
      log1p-def [=>] 74.2%	\[ x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]
      neg-sub0 [=>] 74.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]
      associate-+l- [<=] 74.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]
      neg-sub0 [<=] 74.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]
      neg-mul-1 [=>] 74.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{-1 \cdot y} + y \cdot e^{z}\right)}{t} \]
      *-commutative [=>] 74.2%	\[ x - \frac{\mathsf{log1p}\left(-1 \cdot y + \color{blue}{e^{z} \cdot y}\right)}{t} \]
      distribute-rgt-out [=>] 74.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(-1 + e^{z}\right)}\right)}{t} \]
      +-commutative [=>] 74.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} + -1\right)}\right)}{t} \]
      metadata-eval [<=] 74.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \left(e^{z} + \color{blue}{\left(-1\right)}\right)\right)}{t} \]
      sub-neg [<=] 74.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]
      expm1-def [=>] 96.4%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   if 1.0000006 < (exp.f64 z)

   1. Initial program 47.2%

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

   2. Simplified 47.2%

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

      [Start] 47.2%	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      associate-+l- [=>] 47.2%	\[ x - \frac{\log \color{blue}{\left(1 - \left(y - y \cdot e^{z}\right)\right)}}{t} \]
      sub-neg [=>] 47.2%	\[ x - \frac{\log \color{blue}{\left(1 + \left(-\left(y - y \cdot e^{z}\right)\right)\right)}}{t} \]
      log1p-def [=>] 47.2%	\[ x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]
      neg-sub0 [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]
      associate-+l- [<=] 47.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]
      neg-sub0 [<=] 47.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]
      neg-mul-1 [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{-1 \cdot y} + y \cdot e^{z}\right)}{t} \]
      *-commutative [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(-1 \cdot y + \color{blue}{e^{z} \cdot y}\right)}{t} \]
      distribute-rgt-out [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(-1 + e^{z}\right)}\right)}{t} \]
      +-commutative [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} + -1\right)}\right)}{t} \]
      metadata-eval [<=] 47.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \left(e^{z} + \color{blue}{\left(-1\right)}\right)\right)}{t} \]
      sub-neg [<=] 47.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]
      expm1-def [=>] 47.2%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Taylor expanded in y around inf 47.2%

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

   4. Simplified 47.2%

      \[\leadsto x - \frac{\color{blue}{\log \left(\mathsf{expm1}\left(z\right)\right) + \left(-\left(-\log y\right)\right)}}{t} \]
      Step-by-step derivation

      [Start] 47.2%	\[ x - \frac{\log \left(e^{z} - 1\right) + -1 \cdot \log \left(\frac{1}{y}\right)}{t} \]
      mul-1-neg [=>] 47.2%	\[ x - \frac{\log \left(e^{z} - 1\right) + \color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)}}{t} \]
      expm1-def [=>] 47.2%	\[ x - \frac{\log \color{blue}{\left(\mathsf{expm1}\left(z\right)\right)} + \left(-\log \left(\frac{1}{y}\right)\right)}{t} \]
      log-rec [=>] 47.2%	\[ x - \frac{\log \left(\mathsf{expm1}\left(z\right)\right) + \left(-\color{blue}{\left(-\log y\right)}\right)}{t} \]

   5. Taylor expanded in z around 0 58.6%

      \[\leadsto x - \frac{\color{blue}{\log z + \left(0.5 \cdot z + \log y\right)}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.0000006:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log z + \left(z \cdot 0.5 + \log y\right)}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	91.6%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 1.0000006:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log z + \left(z \cdot 0.5 + \log y\right)}{t}\\ \end{array} \]

Alternative 2
Accuracy	82.5%
Cost	13449

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+35} \lor \neg \left(y \leq 1.6 \cdot 10^{+96}\right):\\ \;\;\;\;\frac{-\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 3
Accuracy	89.9%
Cost	13248

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

Alternative 4
Accuracy	79.5%
Cost	7364

\[\begin{array}{l} \mathbf{if}\;y \leq -1800000000:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 5
Accuracy	76.9%
Cost	6848

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

Alternative 6
Accuracy	64.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+93}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+107}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{z}{t}}{-y}\\ \end{array} \]

Alternative 7
Accuracy	70.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+91}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 8
Accuracy	47.8%
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \end{array} \]

Alternative 9
Accuracy	49.7%
Cost	516

\[\begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-46}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \end{array} \]

Alternative 10
Accuracy	46.5%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023277 
(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)))