Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 14.8 → 2.6

Time: 11.6s

Precision: binary64

Cost: 2248

\[ \begin{array}{c}[x, y] = \mathsf{sort}([x, y])\\ \end{array} \]

Math

\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

↓

\[\begin{array}{l} t_0 := z \cdot \frac{z + 1}{y}\\ t_1 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{z}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t_0}\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ (+ z 1.0) y))) (t_1 (/ (* x y) (* (* z z) (+ z 1.0)))))
   (if (<= t_1 -2e-143)
     (/ (/ x t_0) z)
     (if (<= t_1 5e-270) (/ (* y (/ (/ x z) z)) (+ z 1.0)) (/ (/ x z) t_0)))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

double code(double x, double y, double z) {
	double t_0 = z * ((z + 1.0) / y);
	double t_1 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (t_1 <= -2e-143) {
		tmp = (x / t_0) / z;
	} else if (t_1 <= 5e-270) {
		tmp = (y * ((x / z) / z)) / (z + 1.0);
	} else {
		tmp = (x / z) / t_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 = (x * y) / ((z * z) * (z + 1.0d0))
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) :: t_1
    real(8) :: tmp
    t_0 = z * ((z + 1.0d0) / y)
    t_1 = (x * y) / ((z * z) * (z + 1.0d0))
    if (t_1 <= (-2d-143)) then
        tmp = (x / t_0) / z
    else if (t_1 <= 5d-270) then
        tmp = (y * ((x / z) / z)) / (z + 1.0d0)
    else
        tmp = (x / z) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

↓

public static double code(double x, double y, double z) {
	double t_0 = z * ((z + 1.0) / y);
	double t_1 = (x * y) / ((z * z) * (z + 1.0));
	double tmp;
	if (t_1 <= -2e-143) {
		tmp = (x / t_0) / z;
	} else if (t_1 <= 5e-270) {
		tmp = (y * ((x / z) / z)) / (z + 1.0);
	} else {
		tmp = (x / z) / t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

↓

def code(x, y, z):
	t_0 = z * ((z + 1.0) / y)
	t_1 = (x * y) / ((z * z) * (z + 1.0))
	tmp = 0
	if t_1 <= -2e-143:
		tmp = (x / t_0) / z
	elif t_1 <= 5e-270:
		tmp = (y * ((x / z) / z)) / (z + 1.0)
	else:
		tmp = (x / z) / t_0
	return tmp

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

↓

function code(x, y, z)
	t_0 = Float64(z * Float64(Float64(z + 1.0) / y))
	t_1 = Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-143)
		tmp = Float64(Float64(x / t_0) / z);
	elseif (t_1 <= 5e-270)
		tmp = Float64(Float64(y * Float64(Float64(x / z) / z)) / Float64(z + 1.0));
	else
		tmp = Float64(Float64(x / z) / t_0);
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

↓

function tmp_2 = code(x, y, z)
	t_0 = z * ((z + 1.0) / y);
	t_1 = (x * y) / ((z * z) * (z + 1.0));
	tmp = 0.0;
	if (t_1 <= -2e-143)
		tmp = (x / t_0) / z;
	elseif (t_1 <= 5e-270)
		tmp = (y * ((x / z) / z)) / (z + 1.0);
	else
		tmp = (x / z) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(N[(z + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-143], N[(N[(x / t$95$0), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e-270], N[(N[(y * N[(N[(x / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Target

Original	14.8
Target	3.9
Herbie	2.6

\[\begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < -1.9999999999999999e-143

   1. Initial program 15.0

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   2. Simplified 15.6

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      Proof

      [Start] 15.0	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 15.6	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 3.7

      \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}} \]

   if -1.9999999999999999e-143 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1))) < 4.9999999999999998e-270

   1. Initial program 7.2

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   2. Simplified 2.2

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      Proof

      [Start] 7.2	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 2.2	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 1.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{z}}{z} \cdot y}{z + 1}} \]

   if 4.9999999999999998e-270 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z 1)))

   1. Initial program 28.7

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   2. Simplified 23.3

      \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]
      Proof

      [Start] 28.7	\[ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
      times-frac [=>] 23.3	\[ \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

   3. Applied egg-rr 4.4

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{\frac{z + 1}{y} \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 2.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq -2 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{x}{z \cdot \frac{z + 1}{y}}}{z}\\ \mathbf{elif}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Error	3.7
Cost	2248

\[\begin{array}{l} t_0 := \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{x}}}{z}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{-270}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z + 1}{y}}\\ \end{array} \]

Alternative 2
Error	3.9
Cost	1100

\[\begin{array}{l} t_0 := \frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \mathbf{if}\;z \leq -1 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z}\\ \end{array} \]

Alternative 3
Error	4.1
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{x}\right)}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z}\\ \end{array} \]

Alternative 4
Error	4.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{x}\right)}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z}\\ \end{array} \]

Alternative 5
Error	3.2
Cost	969

\[\begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-109} \lor \neg \left(z \leq 5.8 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 6
Error	16.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.62\right):\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 7
Error	5.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 8
Error	4.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \end{array} \]

Alternative 9
Error	4.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.75\right):\\ \;\;\;\;\frac{y \cdot \frac{\frac{x}{z}}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} - y}{\frac{z}{x}}\\ \end{array} \]

Alternative 10
Error	5.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.86:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 11
Error	5.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{\frac{x}{z}}}\\ \end{array} \]

Alternative 12
Error	5.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{x}{z} \cdot \left(\frac{y}{z} - y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z \cdot z}\\ \end{array} \]

Alternative 13
Error	17.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-108} \lor \neg \left(z \leq 8.6 \cdot 10^{+159}\right):\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 14
Error	17.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-19} \lor \neg \left(z \leq 9.2 \cdot 10^{-27}\right):\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 15
Error	16.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z}{y}}\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]

Alternative 16
Error	18.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{\frac{z}{\frac{y}{z}}}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot z}\\ \end{array} \]

Alternative 17
Error	18.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{z}\\ \end{array} \]

Alternative 18
Error	21.9
Cost	448

\[y \cdot \frac{\frac{x}{z}}{z} \]

Alternative 19
Error	45.6
Cost	384

\[\frac{-y}{\frac{z}{x}} \]

Reproduce

herbie shell --seed 2023023 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))