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

Percentage Accurate: 81.3% → 98.3%

Time: 8.6s

Alternatives: 5

Speedup: 1.1×

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: 81.3% 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: 98.3% 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 82.1%

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

   1. associate-/l* 89.5%

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

   2. associate-*l* 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in z around 0 96.7%

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

   1. +-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. mul-1-neg 96.7%

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

   4. associate-*l/ 98.8%

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

   5. *-commutative 98.8%

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

   6. sub-neg 98.8%

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

7. Simplified 98.8%

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

8. Taylor expanded in y around 0 96.7%

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

   1. *-commutative 96.7%

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

   2. associate-/l* 98.8%

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

10. Simplified 98.8%

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

11. Final simplification 98.8%

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

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{+29} \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.04e+29) (not (<= z 5.2)))
   (- x (/ y z))
   (- x (* (/ z t) -2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.04e+29) || !(z <= 5.2)) {
		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 <= (-1.04d+29)) .or. (.not. (z <= 5.2d0))) 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 <= -1.04e+29) || !(z <= 5.2)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z / t) * -2.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.04e+29) or not (z <= 5.2):
		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 <= -1.04e+29) || !(z <= 5.2))
		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 <= -1.04e+29) || ~((z <= 5.2)))
		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, -1.04e+29], N[Not[LessEqual[z, 5.2]], $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 < -1.0400000000000001e29 or 5.20000000000000018 < z

   1. Initial program 70.6%

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

      1. associate-/l* 86.8%

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

      2. associate-*l* 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in y around 0 91.3%

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

   if -1.0400000000000001e29 < z < 5.20000000000000018

   1. Initial program 93.1%

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

      1. associate-/l* 92.1%

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

      2. associate-*l* 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in y around inf 93.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{+29} \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+129} \lor \neg \left(z \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.1e+129) (not (<= z 1.3e+35))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.1e+129) || !(z <= 1.3e+35)) {
		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 <= (-1.1d+129)) .or. (.not. (z <= 1.3d+35))) 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 <= -1.1e+129) || !(z <= 1.3e+35)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.1e+129) or not (z <= 1.3e+35):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.1e+129) || !(z <= 1.3e+35))
		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 <= -1.1e+129) || ~((z <= 1.3e+35)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.1e+129], N[Not[LessEqual[z, 1.3e+35]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.1e129 or 1.30000000000000003e35 < z

   1. Initial program 65.9%

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

      1. associate-/l* 83.8%

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

      2. associate-*l* 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in y around 0 95.6%

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

   if -1.1e129 < z < 1.30000000000000003e35

   1. Initial program 91.9%

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

      1. associate-/l* 92.9%

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

      2. associate-*l* 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in x around inf 80.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+129} \lor \neg \left(z \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

   1. associate-/l* 89.5%

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

   2. associate-*l* 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in z around 0 96.7%

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

   1. +-commutative 96.7%

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

   2. *-commutative 96.7%

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

   3. mul-1-neg 96.7%

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

   4. associate-*l/ 98.8%

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

   5. *-commutative 98.8%

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

   6. sub-neg 98.8%

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

7. Simplified 98.8%

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

8. Final simplification 98.8%

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

Alternative 5: 74.9% 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 82.1%

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

   1. associate-/l* 89.5%

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

   2. associate-*l* 89.5%

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

3. Simplified 89.5%

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

5. Taylor expanded in x around inf 78.5%

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

6. Final simplification 78.5%

   \[\leadsto x \]
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 2024021 
(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)))))