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

Average Accuracy: 82.3% → 99.9%

Time: 13.6s

Precision: binary64

Cost: 832

Math

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

↓

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

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 (- (/ (* z 2.0) y) (/ 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 / (((z * 2.0) / y) - (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) / (((z * 2.0d0) / y) - (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 / (((z * 2.0) / y) - (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 / (((z * 2.0) / y) - (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(Float64(z * 2.0) / y) - 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 / (((z * 2.0) / y) - (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[(N[(z * 2.0), $MachinePrecision] / y), $MachinePrecision] - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	82.3%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 82.3%

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

2. Simplified 99.9%

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

   [Start] 82.3	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   sub-neg [=>] 82.3	\[ \color{blue}{x + \left(-\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\right)} \]
   associate-/l* [=>] 90.2	\[ x + \left(-\color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right) \]
   *-commutative [=>] 90.2	\[ x + \left(-\frac{\color{blue}{2 \cdot y}}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}\right) \]
   associate-/l* [=>] 90.2	\[ x + \left(-\color{blue}{\frac{2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}}\right) \]
   distribute-neg-frac [=>] 90.2	\[ x + \color{blue}{\frac{-2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}} \]
   metadata-eval [=>] 90.2	\[ x + \frac{\color{blue}{-2}}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}} \]
   associate-/l/ [=>] 82.4	\[ x + \frac{-2}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{y \cdot z}}} \]
   div-sub [=>] 75.1	\[ x + \frac{-2}{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z}{y \cdot z} - \frac{y \cdot t}{y \cdot z}}} \]
   times-frac [=>] 90.1	\[ x + \frac{-2}{\color{blue}{\frac{z \cdot 2}{y} \cdot \frac{z}{z}} - \frac{y \cdot t}{y \cdot z}} \]
   *-inverses [=>] 90.1	\[ x + \frac{-2}{\frac{z \cdot 2}{y} \cdot \color{blue}{1} - \frac{y \cdot t}{y \cdot z}} \]
   *-rgt-identity [=>] 90.1	\[ x + \frac{-2}{\color{blue}{\frac{z \cdot 2}{y}} - \frac{y \cdot t}{y \cdot z}} \]
   *-commutative [=>] 90.1	\[ x + \frac{-2}{\frac{\color{blue}{2 \cdot z}}{y} - \frac{y \cdot t}{y \cdot z}} \]
   associate-*l/ [<=] 90.0	\[ x + \frac{-2}{\color{blue}{\frac{2}{y} \cdot z} - \frac{y \cdot t}{y \cdot z}} \]
   *-commutative [<=] 90.0	\[ x + \frac{-2}{\color{blue}{z \cdot \frac{2}{y}} - \frac{y \cdot t}{y \cdot z}} \]
   times-frac [=>] 99.9	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{y}{y} \cdot \frac{t}{z}}} \]
   *-inverses [=>] 99.9	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{1} \cdot \frac{t}{z}} \]
   *-lft-identity [=>] 99.9	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{t}{z}}} \]

3. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \frac{t}{z}} \]
   associate-*r/ [=>] 99.9	\[ x + \frac{-2}{\color{blue}{\frac{z \cdot 2}{y}} - \frac{t}{z}} \]

4. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	832

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

Alternative 2
Accuracy	81.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-76}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-13}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \end{array} \]

Alternative 3
Accuracy	89.3%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -5500000000:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{z}{y}}\\ \end{array} \]

Alternative 4
Accuracy	81.3%
Cost	585

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-76} \lor \neg \left(z \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	74.1%
Cost	520

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+133}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{+88}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	76.1%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023146 
(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)))))