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

Percentage Accurate: 82.6% → 99.9%

Time: 10.2s

Alternatives: 6

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: 82.6% 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

\[\begin{array}{l} \\ x + \frac{-1}{\frac{-0.5}{\frac{z}{t}} + \frac{z}{y}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (/ -1.0 (+ (/ -0.5 (/ z t)) (/ z y)))))

C

double code(double x, double y, double z, double t) {
	return x + (-1.0 / ((-0.5 / (z / t)) + (z / y)));
}

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) / (((-0.5d0) / (z / t)) + (z / y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (-1.0 / ((-0.5 / (z / t)) + (z / y)));
}

Python

def code(x, y, z, t):
	return x + (-1.0 / ((-0.5 / (z / t)) + (z / y)))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(-1.0 / Float64(Float64(-0.5 / Float64(z / t)) + Float64(z / y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-1.0 / ((-0.5 / (z / t)) + (z / y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(-1.0 / N[(N[(-0.5 / N[(z / t), $MachinePrecision]), $MachinePrecision] + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Simplified 89.1%

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

   1. associate-*r* 89.1%

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

   2. associate-/l* 82.7%

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

   3. clear-num 82.7%

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

   4. *-commutative 82.7%

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

   5. associate-*l* 82.7%

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

   6. pow2 82.7%

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

   7. *-commutative 82.7%

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

   8. *-commutative 82.7%

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

6. Applied egg-rr 82.7%

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

7. Taylor expanded in y around inf 99.9%

   \[\leadsto x - \frac{1}{\color{blue}{-0.5 \cdot \frac{t}{z} + \frac{z}{y}}} \]
8. Step-by-step derivation

   1. clear-num 99.9%

      \[\leadsto x - \frac{1}{-0.5 \cdot \color{blue}{\frac{1}{\frac{z}{t}}} + \frac{z}{y}} \]

   2. un-div-inv 99.9%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5}{\frac{z}{t}}} + \frac{z}{y}} \]

9. Applied egg-rr 99.9%

   \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5}{\frac{z}{t}}} + \frac{z}{y}} \]

10. Final simplification 99.9%

    \[\leadsto x + \frac{-1}{\frac{-0.5}{\frac{z}{t}} + \frac{z}{y}} \]
11. Add Preprocessing

Alternative 2: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+57} \lor \neg \left(z \leq 2.6 \cdot 10^{+67}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e+57) (not (<= z 2.6e+67)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e+57) || !(z <= 2.6e+67)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	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 <= (-3.4d+57)) .or. (.not. (z <= 2.6d+67))) then
        tmp = x - (y / z)
    else
        tmp = x - ((z * (-2.0d0)) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e+57) || !(z <= 2.6e+67)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.4e+57) or not (z <= 2.6e+67):
		tmp = x - (y / z)
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e+57) || !(z <= 2.6e+67))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(Float64(z * -2.0) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.4e+57) || ~((z <= 2.6e+67)))
		tmp = x - (y / z);
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e+57], N[Not[LessEqual[z, 2.6e+67]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.39999999999999992e57 or 2.6e67 < z

   1. Initial program 73.1%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around 0 95.2%

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

   if -3.39999999999999992e57 < z < 2.6e67

   1. Initial program 88.9%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around inf 86.5%

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

      1. associate-*r/ 86.5%

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

      2. *-commutative 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+57} \lor \neg \left(z \leq 2.6 \cdot 10^{+67}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-41} \lor \neg \left(z \leq 1.1 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -8e-41) (not (<= z 1.1e-36))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -8e-41) || !(z <= 1.1e-36)) {
		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 <= (-8d-41)) .or. (.not. (z <= 1.1d-36))) 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 <= -8e-41) || !(z <= 1.1e-36)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -8e-41) or not (z <= 1.1e-36):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -8e-41) || !(z <= 1.1e-36))
		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 <= -8e-41) || ~((z <= 1.1e-36)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -8e-41], N[Not[LessEqual[z, 1.1e-36]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -8.00000000000000005e-41 or 1.1e-36 < z

   1. Initial program 77.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in y around 0 84.7%

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

   if -8.00000000000000005e-41 < z < 1.1e-36

   1. Initial program 89.1%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around inf 69.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{-41} \lor \neg \left(z \leq 1.1 \cdot 10^{-36}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-218}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-244}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.2e-218) x (if (<= x 5e-244) (* z (/ 2.0 t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e-218) {
		tmp = x;
	} else if (x <= 5e-244) {
		tmp = z * (2.0 / t);
	} 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 (x <= (-4.2d-218)) then
        tmp = x
    else if (x <= 5d-244) then
        tmp = z * (2.0d0 / t)
    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 (x <= -4.2e-218) {
		tmp = x;
	} else if (x <= 5e-244) {
		tmp = z * (2.0 / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.2e-218:
		tmp = x
	elif x <= 5e-244:
		tmp = z * (2.0 / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.2e-218)
		tmp = x;
	elseif (x <= 5e-244)
		tmp = Float64(z * Float64(2.0 / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.2e-218)
		tmp = x;
	elseif (x <= 5e-244)
		tmp = z * (2.0 / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.2e-218], x, If[LessEqual[x, 5e-244], N[(z * N[(2.0 / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999988e-218 or 4.99999999999999998e-244 < x

   1. Initial program 85.3%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around inf 78.8%

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

   if -4.19999999999999988e-218 < x < 4.99999999999999998e-244

   1. Initial program 68.1%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in y around inf 62.0%

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

      1. associate-*r/ 62.0%

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

      2. *-commutative 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in x around 0 53.7%

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

      1. associate-*r/ 53.7%

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

      2. *-commutative 53.7%

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

      3. associate-*r/ 53.5%

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

   10. Simplified 53.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-218}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-244}:\\ \;\;\;\;z \cdot \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{\frac{z}{y} + -0.5 \cdot \frac{t}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (/ -1.0 (+ (/ z y) (* -0.5 (/ t z))))))

C

double code(double x, double y, double z, double t) {
	return x + (-1.0 / ((z / y) + (-0.5 * (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 + ((-1.0d0) / ((z / y) + ((-0.5d0) * (t / z))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x + (-1.0 / ((z / y) + (-0.5 * (t / z))));
}

Python

def code(x, y, z, t):
	return x + (-1.0 / ((z / y) + (-0.5 * (t / z))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(-1.0 / Float64(Float64(z / y) + Float64(-0.5 * Float64(t / z)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-1.0 / ((z / y) + (-0.5 * (t / z))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(-1.0 / N[(N[(z / y), $MachinePrecision] + N[(-0.5 * N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Simplified 89.1%

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

   1. associate-*r* 89.1%

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

   2. associate-/l* 82.7%

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

   3. clear-num 82.7%

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

   4. *-commutative 82.7%

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

   5. associate-*l* 82.7%

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

   6. pow2 82.7%

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

   7. *-commutative 82.7%

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

   8. *-commutative 82.7%

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

6. Applied egg-rr 82.7%

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

7. Taylor expanded in y around inf 99.9%

   \[\leadsto x - \frac{1}{\color{blue}{-0.5 \cdot \frac{t}{z} + \frac{z}{y}}} \]

8. Final simplification 99.9%

   \[\leadsto x + \frac{-1}{\frac{z}{y} + -0.5 \cdot \frac{t}{z}} \]
9. Add Preprocessing

Alternative 6: 75.4% 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 82.7%

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

3. Simplified 89.1%

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

5. Taylor expanded in x around inf 69.8%

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

6. Final simplification 69.8%

   \[\leadsto x \]
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 2024078 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))