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

Percentage Accurate: 82.0% → 99.9%

Time: 6.9s

Alternatives: 3

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

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

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)));
}

Python

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

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

MATLAB

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

Initial Program: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

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)));
}

Python

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

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

MATLAB

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{2}{\frac{z}{\frac{y}{2}} - \frac{t}{z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (2.0 / ((z / (y / 2.0)) - (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 - (2.0d0 / ((z / (y / 2.0d0)) - (t / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Step-by-step derivation

   1. associate-/l* 91.3%

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

   2. associate-*l* 91.3%

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

3. Simplified 91.3%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 96.6%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{-1 \cdot \frac{t \cdot y}{z} + 2 \cdot z}} \]
6. Step-by-step derivation

   1. +-commutative 96.6%

      \[\leadsto x - \frac{y \cdot 2}{\color{blue}{2 \cdot z + -1 \cdot \frac{t \cdot y}{z}}} \]

   2. mul-1-neg 96.6%

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

   3. *-commutative 96.6%

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

   4. associate-*r/ 98.8%

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

   5. unsub-neg 98.8%

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

   6. *-commutative 98.8%

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

   7. associate-*r/ 96.6%

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

   8. associate-*l/ 98.1%

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

   9. *-commutative 98.1%

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

7. Simplified 98.1%

   \[\leadsto x - \frac{y \cdot 2}{\color{blue}{z \cdot 2 - t \cdot \frac{y}{z}}} \]
8. Step-by-step derivation

   1. add-log-exp 69.4%

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

   2. *-un-lft-identity 69.4%

      \[\leadsto x - \log \color{blue}{\left(1 \cdot e^{\frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}}\right)} \]

   3. log-prod 69.4%

      \[\leadsto x - \color{blue}{\left(\log 1 + \log \left(e^{\frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}}\right)\right)} \]

   4. metadata-eval 69.4%

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

   5. add-log-exp 98.1%

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

   6. *-commutative 98.1%

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

   7. *-un-lft-identity 98.1%

      \[\leadsto x - \left(0 + \frac{2 \cdot y}{\color{blue}{1 \cdot \left(z \cdot 2 - t \cdot \frac{y}{z}\right)}}\right) \]

   8. times-frac 98.1%

      \[\leadsto x - \left(0 + \color{blue}{\frac{2}{1} \cdot \frac{y}{z \cdot 2 - t \cdot \frac{y}{z}}}\right) \]

   9. metadata-eval 98.1%

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

   10. *-commutative 98.1%

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

9. Applied egg-rr 98.1%

   \[\leadsto x - \color{blue}{\left(0 + 2 \cdot \frac{y}{2 \cdot z - t \cdot \frac{y}{z}}\right)} \]
10. Step-by-step derivation

    1. +-lft-identity 98.1%

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

    2. associate-*r/ 98.1%

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

    3. associate-/l* 98.0%

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

    4. div-sub 98.0%

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

    5. *-commutative 98.0%

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

    6. associate-/l* 98.0%

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

    7. associate-*r/ 96.5%

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

    8. associate-*l/ 98.7%

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

    9. associate-/l* 99.9%

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

    10. *-inverses 99.9%

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

    11. associate-/r* 99.9%

        \[\leadsto x - \frac{2}{\frac{z}{\frac{y}{2}} - \color{blue}{\frac{t}{z \cdot 1}}} \]

    12. *-rgt-identity 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

    \[\leadsto x - \frac{2}{\frac{z}{\frac{y}{2}} - \frac{t}{z}} \]
13. Add Preprocessing

Alternative 2: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -95 \lor \neg \left(z \leq 2.3 \cdot 10^{+52}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -95.0) (not (<= z 2.3e+52)))
   (- x (/ y z))
   (- x (* (/ z t) -2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -95.0) || !(z <= 2.3e+52)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	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
    real(8) :: tmp
    if ((z <= (-95.0d0)) .or. (.not. (z <= 2.3d+52))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z / t) * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -95.0) || !(z <= 2.3e+52)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -95.0) or not (z <= 2.3e+52):
		tmp = x - (y / z)
	else:
		tmp = x - ((z / t) * -2.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -95.0) || !(z <= 2.3e+52))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z / t) * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -95.0) || ~((z <= 2.3e+52)))
		tmp = x - (y / z);
	else
		tmp = x - ((z / t) * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -95.0], N[Not[LessEqual[z, 2.3e+52]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -95 or 2.3e52 < z

   1. Initial program 69.3%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 87.1%

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

      2. associate-*l* 87.1%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 92.0%

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

   if -95 < z < 2.3e52

   1. Initial program 93.8%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
   2. Step-by-step derivation

      1. associate-/l* 94.6%

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

      2. associate-*l* 94.6%

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

   3. Simplified 94.6%

      \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 87.8%

      \[\leadsto x - \color{blue}{-2 \cdot \frac{z}{t}} \]
   6. Step-by-step derivation

      1. *-commutative 87.8%

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

   7. Simplified 87.8%

      \[\leadsto x - \color{blue}{\frac{z}{t} \cdot -2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -95 \lor \neg \left(z \leq 2.3 \cdot 10^{+52}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{z} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (y / 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 / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

   \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
2. Step-by-step derivation

   1. associate-/l* 91.3%

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

   2. associate-*l* 91.3%

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

3. Simplified 91.3%

   \[\leadsto \color{blue}{x - \frac{y \cdot 2}{\frac{z \cdot \left(2 \cdot z\right) - y \cdot t}{z}}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 59.8%

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

6. Final simplification 59.8%

   \[\leadsto x - \frac{y}{z} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / 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 - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Python

def code(x, y, z, t):
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(z / y) - Float64(Float64(t / 2.0) / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / ((z / y) - ((t / 2.0) / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1.0 / N[(N[(z / y), $MachinePrecision] - N[(N[(t / 2.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024041 
(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)))))