Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2

Average Accuracy: 41.7% → 43.8%

Time: 12.0s

Precision: binary64

Cost: 26948

Math

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

↓

\[\begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;\left(x + t_0\right) - z \leq 10^{+22}:\\ \;\;\;\;e^{\left(x - z\right) + 3 \cdot \log \left(\log y \cdot \left(y \cdot 0.3333333333333333\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(t_0 + 1\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (log y))))
   (if (<= (- (+ x t_0) z) 1e+22)
     (exp
      (+ (- x z) (* 3.0 (log (+ (* (log y) (* y 0.3333333333333333)) 1.0)))))
     (* (+ x 1.0) (+ t_0 1.0)))))

C

double code(double x, double y, double z) {
	return exp(((x + (y * log(y))) - z));
}

↓

double code(double x, double y, double z) {
	double t_0 = y * log(y);
	double tmp;
	if (((x + t_0) - z) <= 1e+22) {
		tmp = exp(((x - z) + (3.0 * log(((log(y) * (y * 0.3333333333333333)) + 1.0)))));
	} else {
		tmp = (x + 1.0) * (t_0 + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = exp(((x + (y * log(y))) - z))
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * log(y)
    if (((x + t_0) - z) <= 1d+22) then
        tmp = exp(((x - z) + (3.0d0 * log(((log(y) * (y * 0.3333333333333333d0)) + 1.0d0)))))
    else
        tmp = (x + 1.0d0) * (t_0 + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return Math.exp(((x + (y * Math.log(y))) - z));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = y * Math.log(y);
	double tmp;
	if (((x + t_0) - z) <= 1e+22) {
		tmp = Math.exp(((x - z) + (3.0 * Math.log(((Math.log(y) * (y * 0.3333333333333333)) + 1.0)))));
	} else {
		tmp = (x + 1.0) * (t_0 + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	return math.exp(((x + (y * math.log(y))) - z))

↓

def code(x, y, z):
	t_0 = y * math.log(y)
	tmp = 0
	if ((x + t_0) - z) <= 1e+22:
		tmp = math.exp(((x - z) + (3.0 * math.log(((math.log(y) * (y * 0.3333333333333333)) + 1.0)))))
	else:
		tmp = (x + 1.0) * (t_0 + 1.0)
	return tmp

Julia

function code(x, y, z)
	return exp(Float64(Float64(x + Float64(y * log(y))) - z))
end

↓

function code(x, y, z)
	t_0 = Float64(y * log(y))
	tmp = 0.0
	if (Float64(Float64(x + t_0) - z) <= 1e+22)
		tmp = exp(Float64(Float64(x - z) + Float64(3.0 * log(Float64(Float64(log(y) * Float64(y * 0.3333333333333333)) + 1.0)))));
	else
		tmp = Float64(Float64(x + 1.0) * Float64(t_0 + 1.0));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = exp(((x + (y * log(y))) - z));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = y * log(y);
	tmp = 0.0;
	if (((x + t_0) - z) <= 1e+22)
		tmp = exp(((x - z) + (3.0 * log(((log(y) * (y * 0.3333333333333333)) + 1.0)))));
	else
		tmp = (x + 1.0) * (t_0 + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[Exp[N[(N[(x + N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]], $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(x + t$95$0), $MachinePrecision] - z), $MachinePrecision], 1e+22], N[Exp[N[(N[(x - z), $MachinePrecision] + N[(3.0 * N[Log[N[(N[(N[Log[y], $MachinePrecision] * N[(y * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] * N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]]]

Target

Original	41.7%
Target	41.7%
Herbie	43.8%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z) < 1e22

   1. Initial program 97.1%

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

   2. Simplified 73.7%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
      Proof

      [Start] 97.1	\[ e^{\left(x + y \cdot \log y\right) - z} \]
      +-commutative [=>] 97.1	\[ e^{\color{blue}{\left(y \cdot \log y + x\right)} - z} \]
      associate--l+ [=>] 97.1	\[ e^{\color{blue}{y \cdot \log y + \left(x - z\right)}} \]
      exp-sum [=>] 73.7	\[ \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]
      *-commutative [=>] 73.7	\[ e^{\color{blue}{\log y \cdot y}} \cdot e^{x - z} \]
      exp-to-pow [=>] 73.7	\[ \color{blue}{{y}^{y}} \cdot e^{x - z} \]

   3. Taylor expanded in y around 0 96.4%

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

   4. Simplified 96.4%

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

      [Start] 96.4	\[ y \cdot \left(e^{x - z} \cdot \log y\right) + e^{x - z} \]
      *-commutative [=>] 96.4	\[ y \cdot \color{blue}{\left(\log y \cdot e^{x - z}\right)} + e^{x - z} \]
      associate-*r* [=>] 96.4	\[ \color{blue}{\left(y \cdot \log y\right) \cdot e^{x - z}} + e^{x - z} \]
      distribute-lft1-in [=>] 96.4	\[ \color{blue}{\left(y \cdot \log y + 1\right) \cdot e^{x - z}} \]

   5. Applied egg-rr 73.0%

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

      [Start] 96.4	\[ \left(y \cdot \log y + 1\right) \cdot e^{x - z} \]
      add-exp-log [=>] 96.4	\[ \color{blue}{e^{\log \left(\left(y \cdot \log y + 1\right) \cdot e^{x - z}\right)}} \]
      *-commutative [=>] 96.4	\[ e^{\log \color{blue}{\left(e^{x - z} \cdot \left(y \cdot \log y + 1\right)\right)}} \]
      log-prod [=>] 96.4	\[ e^{\color{blue}{\log \left(e^{x - z}\right) + \log \left(y \cdot \log y + 1\right)}} \]
      add-log-exp [<=] 96.4	\[ e^{\color{blue}{\left(x - z\right)} + \log \left(y \cdot \log y + 1\right)} \]
      +-commutative [=>] 96.4	\[ e^{\left(x - z\right) + \log \color{blue}{\left(1 + y \cdot \log y\right)}} \]
      log1p-udef [<=] 96.4	\[ e^{\left(x - z\right) + \color{blue}{\mathsf{log1p}\left(y \cdot \log y\right)}} \]
      *-rgt-identity [<=] 96.4	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(\color{blue}{\left(y \cdot \log y\right) \cdot 1}\right)} \]
      *-rgt-identity [=>] 96.4	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(\color{blue}{y \cdot \log y}\right)} \]
      log-pow [<=] 73.0	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(\color{blue}{\log \left({y}^{y}\right)}\right)} \]

   6. Simplified 96.4%

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

      [Start] 73.0	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(\log \left({y}^{y}\right)\right)} \]
      log-pow [=>] 96.4	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(\color{blue}{y \cdot \log y}\right)} \]

   7. Applied egg-rr 96.4%

      \[\leadsto e^{\left(x - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)\right)}} \]
      Proof

      [Start] 96.4	\[ e^{\left(x - z\right) + \mathsf{log1p}\left(y \cdot \log y\right)} \]
      add-log-exp [=>] 96.4	\[ e^{\left(x - z\right) + \color{blue}{\log \left(e^{\mathsf{log1p}\left(y \cdot \log y\right)}\right)}} \]
      add-cube-cbrt [=>] 96.4	\[ e^{\left(x - z\right) + \log \color{blue}{\left(\left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)}} \]
      log-prod [=>] 96.4	\[ e^{\left(x - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)}} \]
      log1p-udef [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{e^{\color{blue}{\log \left(1 + y \cdot \log y\right)}}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      add-exp-log [<=] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\color{blue}{1 + y \cdot \log y}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      +-commutative [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\color{blue}{y \cdot \log y + 1}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      fma-def [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(y, \log y, 1\right)}} \cdot \sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      log1p-udef [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{e^{\color{blue}{\log \left(1 + y \cdot \log y\right)}}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      add-exp-log [<=] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{\color{blue}{1 + y \cdot \log y}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      +-commutative [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{\color{blue}{y \cdot \log y + 1}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]
      fma-def [=>] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{\color{blue}{\mathsf{fma}\left(y, \log y, 1\right)}}\right) + \log \left(\sqrt[3]{e^{\mathsf{log1p}\left(y \cdot \log y\right)}}\right)\right)} \]

   8. Simplified 96.4%

      \[\leadsto e^{\left(x - z\right) + \color{blue}{3 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)}} \]
      Proof

      [Start] 96.4	\[ e^{\left(x - z\right) + \left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)\right)} \]
      log-prod [=>] 96.4	\[ e^{\left(x - z\right) + \left(\color{blue}{\left(\log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right) + \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)\right)} \]
      count-2 [=>] 96.4	\[ e^{\left(x - z\right) + \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)} + \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)\right)} \]
      distribute-lft1-in [=>] 96.4	\[ e^{\left(x - z\right) + \color{blue}{\left(2 + 1\right) \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)}} \]
      metadata-eval [=>] 96.4	\[ e^{\left(x - z\right) + \color{blue}{3} \cdot \log \left(\sqrt[3]{\mathsf{fma}\left(y, \log y, 1\right)}\right)} \]

   9. Taylor expanded in y around 0 96.4%

      \[\leadsto e^{\left(x - z\right) + 3 \cdot \log \color{blue}{\left(1 + 0.3333333333333333 \cdot \left(y \cdot \log y\right)\right)}} \]

   10. Simplified 96.4%

       \[\leadsto e^{\left(x - z\right) + 3 \cdot \log \color{blue}{\left(\left(0.3333333333333333 \cdot y\right) \cdot \log y + 1\right)}} \]
       Proof

       [Start] 96.4	\[ e^{\left(x - z\right) + 3 \cdot \log \left(1 + 0.3333333333333333 \cdot \left(y \cdot \log y\right)\right)} \]
       +-commutative [=>] 96.4	\[ e^{\left(x - z\right) + 3 \cdot \log \color{blue}{\left(0.3333333333333333 \cdot \left(y \cdot \log y\right) + 1\right)}} \]
       associate-*r* [=>] 96.4	\[ e^{\left(x - z\right) + 3 \cdot \log \left(\color{blue}{\left(0.3333333333333333 \cdot y\right) \cdot \log y} + 1\right)} \]

   if 1e22 < (-.f64 (+.f64 x (*.f64 y (log.f64 y))) z)

   1. Initial program 0.0%

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

   2. Simplified 0.0%

      \[\leadsto \color{blue}{{y}^{y} \cdot e^{x - z}} \]
      Proof

      [Start] 0.0	\[ e^{\left(x + y \cdot \log y\right) - z} \]
      +-commutative [=>] 0.0	\[ e^{\color{blue}{\left(y \cdot \log y + x\right)} - z} \]
      associate--l+ [=>] 0.0	\[ e^{\color{blue}{y \cdot \log y + \left(x - z\right)}} \]
      exp-sum [=>] 0.0	\[ \color{blue}{e^{y \cdot \log y} \cdot e^{x - z}} \]
      *-commutative [=>] 0.0	\[ e^{\color{blue}{\log y \cdot y}} \cdot e^{x - z} \]
      exp-to-pow [=>] 0.0	\[ \color{blue}{{y}^{y}} \cdot e^{x - z} \]

   3. Taylor expanded in x around 0 0.6%

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

   4. Simplified 0.6%

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

      [Start] 0.6	\[ e^{-z} \cdot {y}^{y} + e^{-z} \cdot \left({y}^{y} \cdot x\right) \]
      distribute-lft-out [=>] 0.6	\[ \color{blue}{e^{-z} \cdot \left({y}^{y} + {y}^{y} \cdot x\right)} \]
      exp-neg [=>] 0.6	\[ \color{blue}{\frac{1}{e^{z}}} \cdot \left({y}^{y} + {y}^{y} \cdot x\right) \]
      associate-*l/ [=>] 0.6	\[ \color{blue}{\frac{1 \cdot \left({y}^{y} + {y}^{y} \cdot x\right)}{e^{z}}} \]
      *-lft-identity [=>] 0.6	\[ \frac{\color{blue}{{y}^{y} + {y}^{y} \cdot x}}{e^{z}} \]
      *-commutative [=>] 0.6	\[ \frac{{y}^{y} + \color{blue}{x \cdot {y}^{y}}}{e^{z}} \]
      distribute-rgt1-in [=>] 0.6	\[ \frac{\color{blue}{\left(x + 1\right) \cdot {y}^{y}}}{e^{z}} \]
      *-commutative [=>] 0.6	\[ \frac{\color{blue}{{y}^{y} \cdot \left(x + 1\right)}}{e^{z}} \]

   5. Applied egg-rr 0.6%

      \[\leadsto \color{blue}{0 + \frac{{y}^{y}}{e^{z - \mathsf{log1p}\left(x\right)}}} \]
      Proof

      [Start] 0.6	\[ \frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}} \]
      add-log-exp [=>] 0.0	\[ \color{blue}{\log \left(e^{\frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}}}\right)} \]
      *-un-lft-identity [=>] 0.0	\[ \log \color{blue}{\left(1 \cdot e^{\frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}}}\right)} \]
      log-prod [=>] 0.0	\[ \color{blue}{\log 1 + \log \left(e^{\frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}}}\right)} \]
      metadata-eval [=>] 0.0	\[ \color{blue}{0} + \log \left(e^{\frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}}}\right) \]
      add-log-exp [<=] 0.6	\[ 0 + \color{blue}{\frac{{y}^{y} \cdot \left(x + 1\right)}{e^{z}}} \]
      associate-/l* [=>] 0.6	\[ 0 + \color{blue}{\frac{{y}^{y}}{\frac{e^{z}}{x + 1}}} \]
      add-exp-log [=>] 0.6	\[ 0 + \frac{{y}^{y}}{\frac{e^{z}}{\color{blue}{e^{\log \left(x + 1\right)}}}} \]
      +-commutative [=>] 0.6	\[ 0 + \frac{{y}^{y}}{\frac{e^{z}}{e^{\log \color{blue}{\left(1 + x\right)}}}} \]
      log1p-udef [<=] 0.6	\[ 0 + \frac{{y}^{y}}{\frac{e^{z}}{e^{\color{blue}{\mathsf{log1p}\left(x\right)}}}} \]
      div-exp [=>] 0.6	\[ 0 + \frac{{y}^{y}}{\color{blue}{e^{z - \mathsf{log1p}\left(x\right)}}} \]

   6. Simplified 0.6%

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

      [Start] 0.6	\[ 0 + \frac{{y}^{y}}{e^{z - \mathsf{log1p}\left(x\right)}} \]
      +-lft-identity [=>] 0.6	\[ \color{blue}{\frac{{y}^{y}}{e^{z - \mathsf{log1p}\left(x\right)}}} \]

   7. Taylor expanded in z around 0 1.4%

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

   8. Simplified 1.4%

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

      [Start] 1.4	\[ \frac{{y}^{y}}{e^{-\log \left(1 + x\right)}} \]
      log1p-def [=>] 1.4	\[ \frac{{y}^{y}}{e^{-\color{blue}{\mathsf{log1p}\left(x\right)}}} \]

   9. Taylor expanded in y around 0 4.1%

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

   10. Simplified 4.1%

       \[\leadsto \color{blue}{\left(1 + x\right) \cdot \left(y \cdot \log y + 1\right)} \]
       Proof

       [Start] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + \frac{1}{e^{-\log \left(1 + x\right)}} \]
       rec-exp [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + \color{blue}{e^{-\left(-\log \left(1 + x\right)\right)}} \]
       log1p-def [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + e^{-\left(-\color{blue}{\mathsf{log1p}\left(x\right)}\right)} \]
       remove-double-neg [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + e^{\color{blue}{\mathsf{log1p}\left(x\right)}} \]
       log1p-def [<=] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + e^{\color{blue}{\log \left(1 + x\right)}} \]
       +-commutative [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + e^{\log \color{blue}{\left(x + 1\right)}} \]
       rem-exp-log [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\log \left(1 + x\right)}} + \color{blue}{\left(x + 1\right)} \]
       log1p-def [=>] 4.1	\[ \frac{y \cdot \log y}{e^{-\color{blue}{\mathsf{log1p}\left(x\right)}}} + \left(x + 1\right) \]
       exp-neg [=>] 4.1	\[ \frac{y \cdot \log y}{\color{blue}{\frac{1}{e^{\mathsf{log1p}\left(x\right)}}}} + \left(x + 1\right) \]
       log1p-def [<=] 4.1	\[ \frac{y \cdot \log y}{\frac{1}{e^{\color{blue}{\log \left(1 + x\right)}}}} + \left(x + 1\right) \]
       +-commutative [=>] 4.1	\[ \frac{y \cdot \log y}{\frac{1}{e^{\log \color{blue}{\left(x + 1\right)}}}} + \left(x + 1\right) \]
       rem-exp-log [=>] 4.1	\[ \frac{y \cdot \log y}{\frac{1}{\color{blue}{x + 1}}} + \left(x + 1\right) \]
       associate-/r/ [=>] 4.1	\[ \color{blue}{\frac{y \cdot \log y}{1} \cdot \left(x + 1\right)} + \left(x + 1\right) \]
       /-rgt-identity [=>] 4.1	\[ \color{blue}{\left(y \cdot \log y\right)} \cdot \left(x + 1\right) + \left(x + 1\right) \]
       *-lft-identity [<=] 4.1	\[ \left(y \cdot \log y\right) \cdot \left(x + 1\right) + \color{blue}{1 \cdot \left(x + 1\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y \cdot \log y\right) - z \leq 10^{+22}:\\ \;\;\;\;e^{\left(x - z\right) + 3 \cdot \log \left(\log y \cdot \left(y \cdot 0.3333333333333333\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(y \cdot \log y + 1\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	43.8%
Cost	20292

\[\begin{array}{l} t_0 := y \cdot \log y\\ t_1 := t_0 + 1\\ \mathbf{if}\;\left(x + t_0\right) - z \leq 2000000:\\ \;\;\;\;t_1 \cdot e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot t_1\\ \end{array} \]

Alternative 2
Accuracy	43.5%
Cost	13892

\[\begin{array}{l} t_0 := y \cdot \log y\\ \mathbf{if}\;\left(x + t_0\right) - z \leq 1:\\ \;\;\;\;e^{x - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) \cdot \left(t_0 + 1\right)\\ \end{array} \]

Alternative 3
Accuracy	41.7%
Cost	6592

\[e^{x - z} \]

Alternative 4
Accuracy	30.3%
Cost	6528

\[e^{-z} \]

Alternative 5
Accuracy	14.5%
Cost	192

\[x + 1 \]

Reproduce

herbie shell --seed 2023153 
(FPCore (x y z)
  :name "Statistics.Distribution.Poisson.Internal:probability from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (exp (+ (- x z) (* (log y) y)))

  (exp (- (+ x (* y (log y))) z)))