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

Average Error: 11.5 → 0.1

Time: 12.7s

Precision: binary64

Cost: 832

Math

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

↓

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

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
 (+ x (/ -2.0 (- (/ 2.0 (/ y z)) (/ t 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) {
	return x + (-2.0 / ((2.0 / (y / z)) - (t / z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (((y * 2.0d0) * z) / (((z * 2.0d0) * z) - (y * t)))
end function

↓

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((-2.0d0) / ((2.0d0 / (y / z)) - (t / z)))
end function

Java

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

↓

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

Python

def code(x, y, z, t):
	return x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)))

↓

def code(x, y, z, t):
	return x + (-2.0 / ((2.0 / (y / z)) - (t / z)))

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)
	return Float64(x + Float64(-2.0 / Float64(Float64(2.0 / Float64(y / z)) - Float64(t / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (((y * 2.0) * z) / (((z * 2.0) * z) - (y * t)));
end

↓

function tmp = code(x, y, z, t)
	tmp = x + (-2.0 / ((2.0 / (y / z)) - (t / z)));
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_] := N[(x + N[(-2.0 / N[(N[(2.0 / N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	11.5
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 11.5

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

2. Simplified 0.1

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

   [Start] 11.5	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   sub-neg [=>] 11.5	\[ \color{blue}{x + \left(-\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\right)} \]
   associate-/l* [=>] 6.5	\[ x + \left(-\color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right) \]
   *-commutative [=>] 6.5	\[ x + \left(-\frac{\color{blue}{2 \cdot y}}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}\right) \]
   associate-/l* [=>] 6.5	\[ x + \left(-\color{blue}{\frac{2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}}\right) \]
   distribute-neg-frac [=>] 6.5	\[ x + \color{blue}{\frac{-2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}} \]
   metadata-eval [=>] 6.5	\[ x + \frac{\color{blue}{-2}}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}} \]
   div-sub [=>] 6.5	\[ x + \frac{-2}{\frac{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z}{z} - \frac{y \cdot t}{z}}}{y}} \]
   div-sub [=>] 6.5	\[ x + \frac{-2}{\color{blue}{\frac{\frac{\left(z \cdot 2\right) \cdot z}{z}}{y} - \frac{\frac{y \cdot t}{z}}{y}}} \]
   associate-/l* [=>] 2.7	\[ x + \frac{-2}{\frac{\color{blue}{\frac{z \cdot 2}{\frac{z}{z}}}}{y} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-/l/ [=>] 2.7	\[ x + \frac{-2}{\color{blue}{\frac{z \cdot 2}{y \cdot \frac{z}{z}}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-inverses [=>] 2.7	\[ x + \frac{-2}{\frac{z \cdot 2}{y \cdot \color{blue}{1}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-rgt-identity [=>] 2.7	\[ x + \frac{-2}{\frac{z \cdot 2}{\color{blue}{y}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-commutative [=>] 2.7	\[ x + \frac{-2}{\frac{\color{blue}{2 \cdot z}}{y} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-*l/ [<=] 2.7	\[ x + \frac{-2}{\color{blue}{\frac{2}{y} \cdot z} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-commutative [<=] 2.7	\[ x + \frac{-2}{\color{blue}{z \cdot \frac{2}{y}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-/l/ [=>] 6.2	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{y \cdot t}{y \cdot z}}} \]
   times-frac [=>] 0.1	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{y}{y} \cdot \frac{t}{z}}} \]
   *-inverses [=>] 0.1	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{1} \cdot \frac{t}{z}} \]
   *-lft-identity [=>] 0.1	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{t}{z}}} \]

3. Applied egg-rr 0.1

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	15.4
Cost	1112

\[\begin{array}{l} t_1 := \frac{2 \cdot z}{t}\\ t_2 := x - \frac{y}{z}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-213}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.46 \cdot 10^{-285}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-292}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-222}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-121}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	832

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

Alternative 3
Error	6.8
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{-20} \lor \neg \left(z \leq 2.3 \cdot 10^{-33}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{-2 \cdot z}{t}\\ \end{array} \]

Alternative 4
Error	11.6
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{-30} \lor \neg \left(z \leq 2.8 \cdot 10^{-31}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	16.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023046 
(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)))))