Average Error: 25.3 → 0.3
Time: 3.6s
Precision: binary64
\[x \cdot \sqrt{y \cdot y - z \cdot z} \]
↓
\[\begin{array}{l} \mathbf{if}\;y \leq -7.238217623544146 \cdot 10^{-284}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, z \cdot \frac{z}{y}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot \frac{-0.5}{\frac{\frac{y}{z}}{z}}\right)\\ \end{array} \]
(FPCore (x y z) :precision binary64 (* x (sqrt (- (* y y) (* z z)))))
↓
(FPCore (x y z)
 :precision binary64
 (if (<= y -7.238217623544146e-284)
   (* x (fma 0.5 (* z (/ z y)) (- y)))
   (fma y x (* x (/ -0.5 (/ (/ y z) z))))))
double code(double x, double y, double z) {
	return x * sqrt(((y * y) - (z * z)));
}
↓
double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.238217623544146e-284) {
		tmp = x * fma(0.5, (z * (z / y)), -y);
	} else {
		tmp = fma(y, x, (x * (-0.5 / ((y / z) / z))));
	}
	return tmp;
}

function code(x, y, z)
	return Float64(x * sqrt(Float64(Float64(y * y) - Float64(z * z))))
end
↓
function code(x, y, z)
	tmp = 0.0
	if (y <= -7.238217623544146e-284)
		tmp = Float64(x * fma(0.5, Float64(z * Float64(z / y)), Float64(-y)));
	else
		tmp = fma(y, x, Float64(x * Float64(-0.5 / Float64(Float64(y / z) / z))));
	end
	return tmp
end

code[x_, y_, z_] := N[(x * N[Sqrt[N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_] := If[LessEqual[y, -7.238217623544146e-284], N[(x * N[(0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] + (-y)), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(x * N[(-0.5 / N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

x \cdot \sqrt{y \cdot y - z \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \leq -7.238217623544146 \cdot 10^{-284}:\\
\;\;\;\;x \cdot \mathsf{fma}\left(0.5, z \cdot \frac{z}{y}, -y\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, x, x \cdot \frac{-0.5}{\frac{\frac{y}{z}}{z}}\right)\\


\end{array}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original25.3
Target0.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y < 2.5816096488251695 \cdot 10^{-278}:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\sqrt{y + z} \cdot \sqrt{y - z}\right)\\ \end{array} \]

Derivation

  1. Split input into 2 regimes

  2. if y < -7.23821762354414562e-284

    1. Initial program 25.2

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

    2. Taylor expanded in y around -inf 3.5

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

    3. Simplified0.3

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

    if -7.23821762354414562e-284 < y

    1. Initial program 25.3

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

    2. Taylor expanded in y around inf 3.9

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.238217623544146 \cdot 10^{-284}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(0.5, z \cdot \frac{z}{y}, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, x \cdot \frac{-0.5}{\frac{\frac{y}{z}}{z}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))