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

Average Error: 11.9 → 3.5

Time: 37.2s

Precision: binary64

Cost: 1224

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+128}:\\ \;\;\;\;\left(x + x\right) - \left(x + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+116}:\\ \;\;\;\;x - y \cdot \frac{z}{z \cdot z - \frac{y \cdot t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

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 (<= z -5.5e+128)
   (- (+ x x) (+ x (/ y z)))
   (if (<= z 1.4e+116)
     (- x (* y (/ z (- (* z z) (/ (* y t) 2.0)))))
     (- x (/ y 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 (z <= -5.5e+128) {
		tmp = (x + x) - (x + (y / z));
	} else if (z <= 1.4e+116) {
		tmp = x - (y * (z / ((z * z) - ((y * t) / 2.0))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (z <= (-5.5d+128)) then
        tmp = (x + x) - (x + (y / z))
    else if (z <= 1.4d+116) then
        tmp = x - (y * (z / ((z * z) - ((y * t) / 2.0d0))))
    else
        tmp = x - (y / z)
    end if
    code = tmp
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) {
	double tmp;
	if (z <= -5.5e+128) {
		tmp = (x + x) - (x + (y / z));
	} else if (z <= 1.4e+116) {
		tmp = x - (y * (z / ((z * z) - ((y * t) / 2.0))));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

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):
	tmp = 0
	if z <= -5.5e+128:
		tmp = (x + x) - (x + (y / z))
	elif z <= 1.4e+116:
		tmp = x - (y * (z / ((z * z) - ((y * t) / 2.0))))
	else:
		tmp = x - (y / 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 (z <= -5.5e+128)
		tmp = Float64(Float64(x + x) - Float64(x + Float64(y / z)));
	elseif (z <= 1.4e+116)
		tmp = Float64(x - Float64(y * Float64(z / Float64(Float64(z * z) - Float64(Float64(y * t) / 2.0)))));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.5e+128)
		tmp = (x + x) - (x + (y / z));
	elseif (z <= 1.4e+116)
		tmp = x - (y * (z / ((z * z) - ((y * t) / 2.0))));
	else
		tmp = x - (y / z);
	end
	tmp_2 = 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[z, -5.5e+128], N[(N[(x + x), $MachinePrecision] - N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e+116], N[(x - N[(y * N[(z / N[(N[(z * z), $MachinePrecision] - N[(N[(y * t), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	11.9
Target	0.1
Herbie	3.5

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

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999998e128

   1. Initial program 26.3

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

   2. Taylor expanded in y around 0 2.1

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{z} + x} \]

   3. Simplified 2.1

      \[\leadsto \color{blue}{x + y \cdot \frac{-1}{z}} \]
      Proof

      [Start] 2.1	\[ -1 \cdot \frac{y}{z} + x \]
      rational_best-simplify-3 [=>] 2.1	\[ \color{blue}{x + -1 \cdot \frac{y}{z}} \]
      rational_best-simplify-55 [=>] 2.1	\[ x + \color{blue}{y \cdot \frac{-1}{z}} \]

   4. Applied egg-rr 2.1

      \[\leadsto \color{blue}{\left(x + x\right) - \left(x + \frac{y}{z}\right)} \]

   if -5.4999999999999998e128 < z < 1.40000000000000002e116

   1. Initial program 5.8

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

   2. Applied egg-rr 3.9

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

   3. Applied egg-rr 3.5

      \[\leadsto x - \color{blue}{\left(z \cdot \frac{-y}{\frac{y \cdot t}{2} - z \cdot z} + 0\right)} \]

   4. Simplified 3.9

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

      [Start] 3.5	\[ x - \left(z \cdot \frac{-y}{\frac{y \cdot t}{2} - z \cdot z} + 0\right) \]
      rational_best-simplify-3 [=>] 3.5	\[ x - \color{blue}{\left(0 + z \cdot \frac{-y}{\frac{y \cdot t}{2} - z \cdot z}\right)} \]
      rational_best-simplify-6 [=>] 3.5	\[ x - \color{blue}{z \cdot \frac{-y}{\frac{y \cdot t}{2} - z \cdot z}} \]
      rational_best-simplify-55 [=>] 3.9	\[ x - \color{blue}{\left(-y\right) \cdot \frac{z}{\frac{y \cdot t}{2} - z \cdot z}} \]
      rational_best-simplify-1 [=>] 3.9	\[ x - \color{blue}{\frac{z}{\frac{y \cdot t}{2} - z \cdot z} \cdot \left(-y\right)} \]
      rational_best-simplify-13 [=>] 3.9	\[ x - \frac{z}{\frac{y \cdot t}{2} - z \cdot z} \cdot \color{blue}{\frac{y}{-1}} \]
      rational_best-simplify-55 [<=] 3.9	\[ x - \color{blue}{y \cdot \frac{\frac{z}{\frac{y \cdot t}{2} - z \cdot z}}{-1}} \]
      rational_best-simplify-53 [=>] 3.9	\[ x - y \cdot \color{blue}{\frac{z}{\left(\frac{y \cdot t}{2} - z \cdot z\right) \cdot -1}} \]
      rational_best-simplify-11 [<=] 3.9	\[ x - y \cdot \frac{z}{\color{blue}{-\left(\frac{y \cdot t}{2} - z \cdot z\right)}} \]
      rational_best-simplify-14 [=>] 3.9	\[ x - y \cdot \frac{z}{\color{blue}{0 - \left(\frac{y \cdot t}{2} - z \cdot z\right)}} \]
      rational_best-simplify-51 [=>] 3.9	\[ x - y \cdot \frac{z}{\color{blue}{z \cdot z - \left(\frac{y \cdot t}{2} - 0\right)}} \]
      rational_best-simplify-9 [=>] 3.9	\[ x - y \cdot \frac{z}{z \cdot z - \color{blue}{\frac{y \cdot t}{2}}} \]

   if 1.40000000000000002e116 < z

   1. Initial program 24.6

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

   2. Taylor expanded in y around 0 3.1

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 3.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+128}:\\ \;\;\;\;\left(x + x\right) - \left(x + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+116}:\\ \;\;\;\;x - y \cdot \frac{z}{z \cdot z - \frac{y \cdot t}{2}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.6
Cost	712

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1950:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.035:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -7.7:\\ \;\;\;\;\left(x + x\right) - \left(x + \frac{y}{z}\right)\\ \mathbf{elif}\;z \leq 16.5:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	12.1
Cost	584

\[\begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -8.6 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.8
Cost	520

\[\begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-192}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{-236}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	15.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023099 
(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)))))