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

Average Error: 7.0 → 1.9

Time: 7.6s

Precision: binary64

Cost: 964

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z \cdot x}}{y}}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 4e+196)
   (/ (/ 1.0 x) (* y (+ (* z z) 1.0)))
   (/ (/ (/ 1.0 (* z x)) y) z)))

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 tmp;
	if ((z * z) <= 4e+196) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} else {
		tmp = ((1.0 / (z * x)) / y) / z;
	}
	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) :: tmp
    if ((z * z) <= 4d+196) then
        tmp = (1.0d0 / x) / (y * ((z * z) + 1.0d0))
    else
        tmp = ((1.0d0 / (z * x)) / y) / z
    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 tmp;
	if ((z * z) <= 4e+196) {
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	} else {
		tmp = ((1.0 / (z * x)) / y) / z;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (z * z) <= 4e+196:
		tmp = (1.0 / x) / (y * ((z * z) + 1.0))
	else:
		tmp = ((1.0 / (z * x)) / y) / z
	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)
	tmp = 0.0
	if (Float64(z * z) <= 4e+196)
		tmp = Float64(Float64(1.0 / x) / Float64(y * Float64(Float64(z * z) + 1.0)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(z * x)) / y) / z);
	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)
	tmp = 0.0;
	if ((z * z) <= 4e+196)
		tmp = (1.0 / x) / (y * ((z * z) + 1.0));
	else
		tmp = ((1.0 / (z * x)) / y) / z;
	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_] := If[LessEqual[N[(z * z), $MachinePrecision], 4e+196], N[(N[(1.0 / x), $MachinePrecision] / N[(y * N[(N[(z * z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	7.0
Target	5.4
Herbie	1.9

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 3.9999999999999998e196

   1. Initial program 1.5

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

   if 3.9999999999999998e196 < (*.f64 z z)

   1. Initial program 16.5

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

   2. Simplified 16.6

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left(y \cdot \mathsf{fma}\left(z, z, 1\right)\right)}} \]
      Proof
      (/.f64 1 (*.f64 x (*.f64 y (fma.f64 z z 1)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 x (*.f64 y (Rewrite<= fma-def_binary64 (+.f64 (*.f64 z z) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 x (*.f64 y (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 z z)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-/r*_binary64 (/.f64 (/.f64 1 x) (*.f64 y (+.f64 1 (*.f64 z z))))): 31 points increase in error, 39 points decrease in error

   3. Taylor expanded in z around inf 16.5

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

   4. Simplified 2.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{z \cdot x}}{y}}{z}} \]
      Proof
      (/.f64 (/.f64 (/.f64 1 (*.f64 z x)) y) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 (*.f64 z x) y))) z): 24 points increase in error, 29 points decrease in error
      (/.f64 (/.f64 1 (Rewrite<= associate-*r*_binary64 (*.f64 z (*.f64 x y)))) z): 22 points increase in error, 38 points decrease in error
      (/.f64 (/.f64 1 (*.f64 z (Rewrite<= *-commutative_binary64 (*.f64 y x)))) z): 0 points increase in error, 0 points decrease in error
      (/.f64 (/.f64 1 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 y x) z))) z): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 (*.f64 (*.f64 y x) z) z))): 17 points increase in error, 26 points decrease in error
      (/.f64 1 (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 y x) (*.f64 z z)))): 42 points increase in error, 11 points decrease in error
      (/.f64 1 (Rewrite=> associate-*l*_binary64 (*.f64 y (*.f64 x (*.f64 z z))))): 34 points increase in error, 15 points decrease in error
      (/.f64 1 (*.f64 y (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 z z) x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 1 (*.f64 y (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) x))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \leq 4 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot \left(z \cdot z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z \cdot x}}{y}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{1}{x \cdot \left(y + \left(z \cdot z\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{z \cdot x}}{y}}{z}\\ \end{array} \]

Alternative 2
Error	5.0
Cost	840

\[\begin{array}{l} t_0 := \frac{1}{y \cdot \left(z \cdot \left(z \cdot x\right)\right)}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.0
Cost	840

\[\begin{array}{l} t_0 := \frac{1}{z \cdot \left(y \cdot \left(z \cdot x\right)\right)}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	2.9
Cost	840

\[\begin{array}{l} t_0 := \frac{\frac{\frac{1}{z \cdot x}}{y}}{z}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{1}{y}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	29.2
Cost	320

\[\frac{1}{x \cdot y} \]

Alternative 6
Error	29.1
Cost	320

\[\frac{\frac{1}{x}}{y} \]

Alternative 7
Error	29.2
Cost	320

\[\frac{\frac{1}{y}}{x} \]

Reproduce

herbie shell --seed 2022329 
(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)))))