Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2

Average Error: 6.5 → 2.1

Time: 23.7s

Precision: binary64

Cost: 968

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{1}{y}}{\left(x \cdot z\right) \cdot z + x}\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-239}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot z\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ 1.0 y) (+ (* (* x z) z) x))))
   (if (<= y -2.3e-137)
     t_0
     (if (<= y 2.1e-239) (/ (/ 1.0 x) (+ (* (* y z) z) y)) t_0))))

C

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

↓

double code(double x, double y, double z) {
	double t_0 = (1.0 / y) / (((x * z) * z) + x);
	double tmp;
	if (y <= -2.3e-137) {
		tmp = t_0;
	} else if (y <= 2.1e-239) {
		tmp = (1.0 / x) / (((y * z) * z) + y);
	} else {
		tmp = 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 = (1.0d0 / x) / (y * (1.0d0 + (z * 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 = (1.0d0 / y) / (((x * z) * z) + x)
    if (y <= (-2.3d-137)) then
        tmp = t_0
    else if (y <= 2.1d-239) then
        tmp = (1.0d0 / x) / (((y * z) * z) + y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (1.0 / y) / (((x * z) * z) + x);
	double tmp;
	if (y <= -2.3e-137) {
		tmp = t_0;
	} else if (y <= 2.1e-239) {
		tmp = (1.0 / x) / (((y * z) * z) + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (1.0 / y) / (((x * z) * z) + x)
	tmp = 0
	if y <= -2.3e-137:
		tmp = t_0
	elif y <= 2.1e-239:
		tmp = (1.0 / x) / (((y * z) * z) + y)
	else:
		tmp = t_0
	return tmp

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(1.0 / y) / Float64(Float64(Float64(x * z) * z) + x))
	tmp = 0.0
	if (y <= -2.3e-137)
		tmp = t_0;
	elseif (y <= 2.1e-239)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(Float64(y * z) * z) + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (1.0 / y) / (((x * z) * z) + x);
	tmp = 0.0;
	if (y <= -2.3e-137)
		tmp = t_0;
	elseif (y <= 2.1e-239)
		tmp = (1.0 / x) / (((y * z) * z) + y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] / N[(N[(N[(x * z), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.3e-137], t$95$0, If[LessEqual[y, 2.1e-239], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Target

Original	6.5
Target	5.9
Herbie	2.1

\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) < -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) < 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -2.30000000000000008e-137 or 2.1000000000000002e-239 < y

   1. Initial program 5.4

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

   2. Applied egg-rr 4.0

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

   3. Taylor expanded in y around 0 5.0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(\left({z}^{2} + 1\right) \cdot x\right)}} \]

   4. Simplified 4.7

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

   5. Applied egg-rr 2.1

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

   if -2.30000000000000008e-137 < y < 2.1000000000000002e-239

   1. Initial program 12.4

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

   2. Applied egg-rr 2.1

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\left(y \cdot z\right) \cdot z + y}} \]
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	1.9
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{y \cdot \left(z \cdot \left(x \cdot z\right) + x\right)}\\ \mathbf{if}\;y \leq -2 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1}{x \cdot \left(z \cdot \left(z \cdot y\right) + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.8
Cost	832

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

Alternative 3
Error	1.9
Cost	832

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

Alternative 4
Error	1.9
Cost	832

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

Alternative 5
Error	4.0
Cost	704

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

Alternative 6
Error	17.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;9 - 9\\ \mathbf{elif}\;z \leq 520000000:\\ \;\;\;\;\frac{1}{y \cdot x}\\ \mathbf{else}:\\ \;\;\;\;9 - 9\\ \end{array} \]

Alternative 7
Error	16.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;9 - 9\\ \mathbf{elif}\;z \leq 700000000:\\ \;\;\;\;\frac{\frac{1}{x}}{y}\\ \mathbf{else}:\\ \;\;\;\;9 - 9\\ \end{array} \]

Alternative 8
Error	16.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -9.8 \cdot 10^{+88}:\\ \;\;\;\;9 - 9\\ \mathbf{elif}\;z \leq 410000000:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;9 - 9\\ \end{array} \]

Alternative 9
Error	41.0
Cost	192

\[9 - 9 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) (- INFINITY)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

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