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

Average Error: 25.0 → 0.5

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -6.9 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \left(t_0 \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-0.5, t_0, y\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (/ y z))))
   (if (<= y -6.9e-271) (- (* 0.5 (* t_0 x)) (* y x)) (* x (fma -0.5 t_0 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 / (y / z);
	double tmp;
	if (y <= -6.9e-271) {
		tmp = (0.5 * (t_0 * x)) - (y * x);
	} else {
		tmp = x * fma(-0.5, t_0, 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(y / z))
	tmp = 0.0
	if (y <= -6.9e-271)
		tmp = Float64(Float64(0.5 * Float64(t_0 * x)) - Float64(y * x));
	else
		tmp = Float64(x * fma(-0.5, t_0, 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[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.9e-271], N[(N[(0.5 * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(x * N[(-0.5 * t$95$0 + y), $MachinePrecision]), $MachinePrecision]]]

Target

Original	25.0
Target	0.5
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if y < -6.9e-271

   1. Initial program 25.1

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

   2. Simplified 25.1

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

   3. Taylor expanded in y around -inf 3.8

      \[\leadsto \color{blue}{0.5 \cdot \frac{{z}^{2} \cdot x}{y} + -1 \cdot \left(y \cdot x\right)} \]

   4. Simplified 0.2

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

   if -6.9e-271 < y

   1. Initial program 24.8

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

   2. Simplified 24.8

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

   3. Taylor expanded in y around inf 3.7

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

   4. Simplified 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.9 \cdot 10^{-271}:\\ \;\;\;\;0.5 \cdot \left(\frac{z}{\frac{y}{z}} \cdot x\right) - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-0.5, \frac{z}{\frac{y}{z}}, y\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(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)))))