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

Average Accuracy: 81.6% → 99.9%

Time: 12.4s

Precision: binary64

Cost: 832

Math

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

↓

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

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

Target

Original	81.6%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 81.6%

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

2. Simplified 99.9%

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

   [Start] 81.6	\[ x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   sub-neg [=>] 81.6	\[ \color{blue}{x + \left(-\frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t}\right)} \]
   associate-/l* [=>] 89.4	\[ x + \left(-\color{blue}{\frac{y \cdot 2}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}}\right) \]
   *-commutative [=>] 89.4	\[ x + \left(-\frac{\color{blue}{2 \cdot y}}{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}\right) \]
   associate-/l* [=>] 89.4	\[ x + \left(-\color{blue}{\frac{2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}}\right) \]
   distribute-neg-frac [=>] 89.4	\[ x + \color{blue}{\frac{-2}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}}} \]
   metadata-eval [=>] 89.4	\[ x + \frac{\color{blue}{-2}}{\frac{\frac{\left(z \cdot 2\right) \cdot z - y \cdot t}{z}}{y}} \]
   div-sub [=>] 89.4	\[ x + \frac{-2}{\frac{\color{blue}{\frac{\left(z \cdot 2\right) \cdot z}{z} - \frac{y \cdot t}{z}}}{y}} \]
   div-sub [=>] 89.4	\[ x + \frac{-2}{\color{blue}{\frac{\frac{\left(z \cdot 2\right) \cdot z}{z}}{y} - \frac{\frac{y \cdot t}{z}}{y}}} \]
   associate-/l* [=>] 95.2	\[ x + \frac{-2}{\frac{\color{blue}{\frac{z \cdot 2}{\frac{z}{z}}}}{y} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-/l/ [=>] 95.2	\[ x + \frac{-2}{\color{blue}{\frac{z \cdot 2}{y \cdot \frac{z}{z}}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-inverses [=>] 95.2	\[ x + \frac{-2}{\frac{z \cdot 2}{y \cdot \color{blue}{1}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-rgt-identity [=>] 95.2	\[ x + \frac{-2}{\frac{z \cdot 2}{\color{blue}{y}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-commutative [=>] 95.2	\[ x + \frac{-2}{\frac{\color{blue}{2 \cdot z}}{y} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-*l/ [<=] 95.2	\[ x + \frac{-2}{\color{blue}{\frac{2}{y} \cdot z} - \frac{\frac{y \cdot t}{z}}{y}} \]
   *-commutative [<=] 95.2	\[ x + \frac{-2}{\color{blue}{z \cdot \frac{2}{y}} - \frac{\frac{y \cdot t}{z}}{y}} \]
   associate-/l/ [=>] 90.2	\[ x + \frac{-2}{z \cdot \frac{2}{y} - \color{blue}{\frac{y \cdot t}{y \cdot z}}} \]
   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. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	76.3%
Cost	848

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;x \leq -2.06 \cdot 10^{-100}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.82 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Accuracy	76.5%
Cost	848

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{-104}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-248}:\\ \;\;\;\;\frac{2}{\frac{t}{z}}\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Accuracy	76.4%
Cost	848

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{-103}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-303}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-247}:\\ \;\;\;\;\frac{z \cdot 2}{t}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	72.5%
Cost	784

\[\begin{array}{l} t_1 := \frac{-y}{z}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-304}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.62 \cdot 10^{-247}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	89.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+44} \lor \neg \left(z \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{2}{t}\\ \end{array} \]

Alternative 6
Accuracy	89.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{+42} \lor \neg \left(z \leq 7 \cdot 10^{-8}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z \cdot 2}{t}\\ \end{array} \]

Alternative 7
Accuracy	72.7%
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-184}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-117}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	75.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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)))))