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

Percentage Accurate: 81.9% → 98.5%

Time: 5.4s

Alternatives: 4

Speedup: 1.5×

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: 81.9% 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: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. associate-/l* 89.1%

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

   2. associate-*l* 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in z around 0 94.5%

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

   1. +-commutative 94.5%

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

   2. *-commutative 94.5%

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

   3. mul-1-neg 94.5%

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

   4. *-commutative 94.5%

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

   5. associate-/l* 96.7%

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

   6. associate-/r/ 95.6%

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

   7. distribute-lft-neg-in 95.6%

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

   8. cancel-sign-sub-inv 95.6%

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

   9. associate-/r/ 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 2: 89.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+35} \lor \neg \left(z \leq 2.3 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9e+35) (not (<= z 2.3e+20)))
   (- x (/ y z))
   (- x (* (/ z t) -2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9e+35) || !(z <= 2.3e+20)) {
		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 <= (-9d+35)) .or. (.not. (z <= 2.3d+20))) 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 <= -9e+35) || !(z <= 2.3e+20)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9e+35) or not (z <= 2.3e+20):
		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 <= -9e+35) || !(z <= 2.3e+20))
		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 <= -9e+35) || ~((z <= 2.3e+20)))
		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, -9e+35], N[Not[LessEqual[z, 2.3e+20]], $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 < -8.9999999999999993e35 or 2.3e20 < z

   1. Initial program 67.5%

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

      1. associate-/l* 87.7%

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

      2. associate-*l* 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in y around 0 88.8%

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

   if -8.9999999999999993e35 < z < 2.3e20

   1. Initial program 92.6%

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

      1. associate-/l* 90.3%

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

      2. associate-*l* 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around inf 92.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+35} \lor \neg \left(z \leq 2.3 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \]

Alternative 3: 80.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{+115} \lor \neg \left(z \leq 1.1 \cdot 10^{-49}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.9e+115) (not (<= z 1.1e-49))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.9e+115) || !(z <= 1.1e-49)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	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 <= (-6.9d+115)) .or. (.not. (z <= 1.1d-49))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.9e+115) || !(z <= 1.1e-49)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6.9e+115) or not (z <= 1.1e-49):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.9e+115) || !(z <= 1.1e-49))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -6.9e+115) || ~((z <= 1.1e-49)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.9e+115], N[Not[LessEqual[z, 1.1e-49]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -6.89999999999999967e115 or 1.09999999999999995e-49 < z

   1. Initial program 67.8%

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

      1. associate-/l* 86.3%

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

      2. associate-*l* 86.3%

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

   3. Simplified 86.3%

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

   4. Taylor expanded in y around 0 88.3%

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

   if -6.89999999999999967e115 < z < 1.09999999999999995e-49

   1. Initial program 91.4%

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

      1. associate-/l* 91.4%

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

      2. associate-*l* 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around inf 82.0%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.9 \cdot 10^{+115} \lor \neg \left(z \leq 1.1 \cdot 10^{-49}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4: 74.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x;
}

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
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 80.9%

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

   1. associate-/l* 89.1%

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

   2. associate-*l* 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in x around inf 76.8%

   \[\leadsto \color{blue}{x} \]

5. Final simplification 76.8%

   \[\leadsto x \]

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 2023290 
(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)))))