Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Average Error: 11.6 → 1.5

Time: 4.1s

Precision: binary64

Math

\[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

↓

\[\begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 8.476544776836846 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{y}{z - \frac{y \cdot t}{2 \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(t \cdot \frac{y}{z}, -0.5, z\right)}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<=
      (- x (/ (* (* y 2.0) z) (- (* z (* 2.0 z)) (* y t))))
      8.476544776836846e+26)
   (- x (/ y (- z (/ (* y t) (* 2.0 z)))))
   (- x (/ y (fma (* t (/ y z)) -0.5 z)))))

C

double code(double x, double y, double z, double t) {
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - (((y * 2.0) * z) / ((z * (2.0 * z)) - (y * t)))) <= 8.476544776836846e+26) {
		tmp = x - (y / (z - ((y * t) / (2.0 * z))));
	} else {
		tmp = x - (y / fma((t * (y / z)), -0.5, z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(Float64(z * 2.0) * z) - Float64(y * t))))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - Float64(Float64(Float64(y * 2.0) * z) / Float64(Float64(z * Float64(2.0 * z)) - Float64(y * t)))) <= 8.476544776836846e+26)
		tmp = Float64(x - Float64(y / Float64(z - Float64(Float64(y * t) / Float64(2.0 * z)))));
	else
		tmp = Float64(x - Float64(y / fma(Float64(t * Float64(y / z)), -0.5, z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(N[(z * 2.0), $MachinePrecision] * z), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(x - N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / N[(N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 8.476544776836846e+26], N[(x - N[(y / N[(z - N[(N[(y * t), $MachinePrecision] / N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision] * -0.5 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	11.6
Target	0.1
Herbie	1.5

\[x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))) < 8.4765447768368462e26

   1. Initial program 3.6

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

   2. Simplified 1.6

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

   if 8.4765447768368462e26 < (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))))

   1. Initial program 25.8

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]

   2. Simplified 5.1

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

   3. Taylor expanded in x around 0 5.1

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

   4. Simplified 1.5

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 8.476544776836846 \cdot 10^{+26}:\\ \;\;\;\;x - \frac{y}{z - \frac{y \cdot t}{2 \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\mathsf{fma}\left(t \cdot \frac{y}{z}, -0.5, z\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022134 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))