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

Percentage Accurate: 56.4% → 90.5%

Time: 19.4s

Precision: binary64

Cost: 19780

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 1.025:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

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) 1.025)
   (- x (/ (log1p (* y (expm1 z))) t))
   (- x (/ y (/ t z)))))

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) <= 1.025) {
		tmp = x - (log1p((y * expm1(z))) / t);
	} else {
		tmp = x - (y / (t / z));
	}
	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) <= 1.025) {
		tmp = x - (Math.log1p((y * Math.expm1(z))) / t);
	} else {
		tmp = x - (y / (t / z));
	}
	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) <= 1.025:
		tmp = x - (math.log1p((y * math.expm1(z))) / t)
	else:
		tmp = x - (y / (t / z))
	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) <= 1.025)
		tmp = Float64(x - Float64(log1p(Float64(y * expm1(z))) / t));
	else
		tmp = Float64(x - Float64(y / Float64(t / z)));
	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], 1.025], N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	56.4%
Target	68.9%
Herbie	90.5%

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 1.0249999999999999

   1. Initial program 62.2%

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

   2. Simplified 98.6%

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

      [Start] 62.2%	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      associate-+l- [=>] 79.6%	\[ x - \frac{\log \color{blue}{\left(1 - \left(y - y \cdot e^{z}\right)\right)}}{t} \]
      sub-neg [=>] 79.6%	\[ x - \frac{\log \color{blue}{\left(1 + \left(-\left(y - y \cdot e^{z}\right)\right)\right)}}{t} \]
      log1p-def [=>] 86.3%	\[ x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]
      neg-sub0 [=>] 86.3%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]
      associate-+l- [<=] 86.3%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]
      neg-sub0 [<=] 86.3%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]
      neg-mul-1 [=>] 86.3%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{-1 \cdot y} + y \cdot e^{z}\right)}{t} \]
      *-commutative [=>] 86.3%	\[ x - \frac{\mathsf{log1p}\left(-1 \cdot y + \color{blue}{e^{z} \cdot y}\right)}{t} \]
      distribute-rgt-out [=>] 86.3%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(-1 + e^{z}\right)}\right)}{t} \]
      +-commutative [=>] 86.3%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} + -1\right)}\right)}{t} \]
      metadata-eval [<=] 86.3%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \left(e^{z} + \color{blue}{\left(-1\right)}\right)\right)}{t} \]
      sub-neg [<=] 86.3%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]
      expm1-def [=>] 98.6%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   if 1.0249999999999999 < (exp.f64 z)

   1. Initial program 34.6%

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

   2. Simplified 34.6%

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

      [Start] 34.6%	\[ x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
      associate-+l- [=>] 34.6%	\[ x - \frac{\log \color{blue}{\left(1 - \left(y - y \cdot e^{z}\right)\right)}}{t} \]
      sub-neg [=>] 34.6%	\[ x - \frac{\log \color{blue}{\left(1 + \left(-\left(y - y \cdot e^{z}\right)\right)\right)}}{t} \]
      log1p-def [=>] 34.6%	\[ x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]
      neg-sub0 [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]
      associate-+l- [<=] 34.6%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]
      neg-sub0 [<=] 34.6%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]
      neg-mul-1 [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{-1 \cdot y} + y \cdot e^{z}\right)}{t} \]
      *-commutative [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(-1 \cdot y + \color{blue}{e^{z} \cdot y}\right)}{t} \]
      distribute-rgt-out [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(-1 + e^{z}\right)}\right)}{t} \]
      +-commutative [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} + -1\right)}\right)}{t} \]
      metadata-eval [<=] 34.6%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \left(e^{z} + \color{blue}{\left(-1\right)}\right)\right)}{t} \]
      sub-neg [<=] 34.6%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\left(e^{z} - 1\right)}\right)}{t} \]
      expm1-def [=>] 34.6%	\[ x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Taylor expanded in z around 0 45.7%

      \[\leadsto x - \color{blue}{\frac{y \cdot z}{t}} \]

   4. Simplified 51.3%

      \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{z}}} \]
      Step-by-step derivation

      [Start] 45.7%	\[ x - \frac{y \cdot z}{t} \]
      associate-/l* [=>] 51.3%	\[ x - \color{blue}{\frac{y}{\frac{t}{z}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 1.025:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.5%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 1.025:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 2
Accuracy	80.7%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	78.3%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+28}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	79.8%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	66.4%
Cost	648

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-203}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.85 \cdot 10^{-177}:\\ \;\;\;\;-\frac{y}{\frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	73.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 7
Accuracy	76.8%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 8
Accuracy	65.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(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)))