Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B

Average Error: 25.1 → 0.3

Time: 2.7s

Precision: binary64

Math

\[x \cdot \sqrt{y \cdot y - z \cdot z} \]

↓

\[\begin{array}{l} t_0 := z \cdot \frac{z}{y}\\ \mathbf{if}\;y \leq -8.509366001875281 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, t_0, -y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(t_0, -0.5, y\right)\\ \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ z y))))
   (if (<= y -8.509366001875281e-289)
     (* x (fma 0.5 t_0 (- y)))
     (* x (fma t_0 -0.5 y)))))

C

double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}

↓

double code(double x, double y, double z) {
	double t_0 = z * (z / y);
	double tmp;
	if (y <= -8.509366001875281e-289) {
		tmp = x * fma(0.5, t_0, -y);
	} else {
		tmp = x * fma(t_0, -0.5, y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(x * sqrt(Float64(Float64(y * y) - Float64(z * z))))
end

↓

function code(x, y, z)
	t_0 = Float64(z * Float64(z / y))
	tmp = 0.0
	if (y <= -8.509366001875281e-289)
		tmp = Float64(x * fma(0.5, t_0, Float64(-y)));
	else
		tmp = Float64(x * fma(t_0, -0.5, y));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.509366001875281e-289], N[(x * N[(0.5 * t$95$0 + (-y)), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$0 * -0.5 + y), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	25.1
Target	0.6
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -8.509366001875281e-289

   1. Initial program 25.3

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Taylor expanded in y around -inf 3.3

      \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{{z}^{2}}{y} - y\right)} \]

   3. Simplified 0.2

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(0.5, \frac{z}{y} \cdot z, -y\right)} \]

   if -8.509366001875281e-289 < y

   1. Initial program 24.8

      \[x \cdot \sqrt{y \cdot y - z \cdot z} \]

   2. Taylor expanded in y around inf 3.6

      \[\leadsto x \cdot \color{blue}{\left(y - 0.5 \cdot \frac{{z}^{2}}{y}\right)} \]

   3. Simplified 0.4

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{z}{y} \cdot z, -0.5, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.509366001875281 \cdot 10^{-289}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, z \cdot \frac{z}{y}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(z \cdot \frac{z}{y}, -0.5, y\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, B"
  :precision binary64

  :herbie-target
  (if (< y 2.5816096488251695e-278) (- (* x y)) (* x (* (sqrt (+ y z)) (sqrt (- y z)))))

  (* x (sqrt (- (* y y) (* z z)))))