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

Average Error: 14.3 → 4.5

Time: 3.8s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ t_1 := \mathsf{hypot}\left(z, \sqrt{z}\right)\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{\frac{1}{y}}\\ \mathbf{elif}\;t_0 \leq 10^{+300}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{t_1}}{t_1}\\ \end{array} \]

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) (+ z 1.0))) (t_1 (hypot z (sqrt z))))
   (if (<= t_0 0.0)
     (* (/ 1.0 z) (/ (/ x (fma z z z)) (/ 1.0 y)))
     (if (<= t_0 1e+300)
       (/ x (* (fma z z z) (/ z y)))
       (/ (/ (* x (/ y z)) t_1) t_1)))))

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) * (z + 1.0);
	double t_1 = hypot(z, sqrt(z));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (1.0 / z) * ((x / fma(z, z, z)) / (1.0 / y));
	} else if (t_0 <= 1e+300) {
		tmp = x / (fma(z, z, z) * (z / y));
	} else {
		tmp = ((x * (y / z)) / t_1) / t_1;
	}
	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(Float64(z * z) * Float64(z + 1.0))
	t_1 = hypot(z, sqrt(z))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(x / fma(z, z, z)) / Float64(1.0 / y)));
	elseif (t_0 <= 1e+300)
		tmp = Float64(x / Float64(fma(z, z, z) * Float64(z / y)));
	else
		tmp = Float64(Float64(Float64(x * Float64(y / z)) / t_1) / t_1);
	end
	return 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[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[z ^ 2 + N[Sqrt[z], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(x / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+300], N[(x / N[(N[(z * z + z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	14.3
Target	4.2
Herbie	4.5

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z 1)) < 0.0

   1. Initial program 21.7

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

   2. Simplified 5.0

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]

   3. Applied egg-rr 5.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{y}}} \]

   4. Applied egg-rr 4.7

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z 1)) < 1.0000000000000001e300

   1. Initial program 6.5

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

   2. Simplified 5.4

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]

   3. Applied egg-rr 5.5

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

   4. Applied egg-rr 6.5

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

   if 1.0000000000000001e300 < (*.f64 (*.f64 z z) (+.f64 z 1))

   1. Initial program 13.2

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

   2. Simplified 5.6

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]

   3. Applied egg-rr 1.1

      \[\leadsto \color{blue}{\frac{\frac{x \cdot \frac{y}{z}}{\mathsf{hypot}\left(z, \sqrt{z}\right)}}{\mathsf{hypot}\left(z, \sqrt{z}\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{\frac{1}{y}}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{+300}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{z}}{\mathsf{hypot}\left(z, \sqrt{z}\right)}}{\mathsf{hypot}\left(z, \sqrt{z}\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))