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

Percentage Accurate: 82.5% → 98.5%

Time: 11.0s

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: 82.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (y * ((t / z) - ((z / y) * 2.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-/l* 86.3%

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

   2. associate-*r* 86.3%

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

   3. associate-*l* 86.3%

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

   4. *-commutative 86.3%

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

   5. clear-num 86.0%

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

   6. un-div-inv 86.0%

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

   7. *-commutative 86.0%

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

   8. associate-*l* 85.9%

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

   9. pow2 85.9%

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

4. Applied egg-rr 85.9%

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

5. Taylor expanded in y around inf 97.1%

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

   1. +-commutative 97.1%

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

   2. mul-1-neg 97.1%

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

   3. unsub-neg 97.1%

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

7. Simplified 97.1%

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

8. Final simplification 97.1%

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

Alternative 2: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+19} \lor \neg \left(z \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.2e+19) (not (<= z 6e+38)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.2e+19) || !(z <= 6e+38)) {
		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 <= (-7.2d+19)) .or. (.not. (z <= 6d+38))) 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 <= -7.2e+19) || !(z <= 6e+38)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.2e+19) or not (z <= 6e+38):
		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 <= -7.2e+19) || !(z <= 6e+38))
		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 <= -7.2e+19) || ~((z <= 6e+38)))
		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, -7.2e+19], N[Not[LessEqual[z, 6e+38]], $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 < -7.2e19 or 6.0000000000000002e38 < z

   1. Initial program 69.1%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around 0 95.0%

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

   if -7.2e19 < z < 6.0000000000000002e38

   1. Initial program 88.5%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around inf 88.4%

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

      1. associate-*r/ 88.4%

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

      2. *-commutative 88.4%

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

   7. Simplified 88.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+19} \lor \neg \left(z \leq 6 \cdot 10^{+38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -115000 \lor \neg \left(z \leq 2.9 \cdot 10^{+38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -115000.0) (not (<= z 2.9e+38))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -115000.0) || !(z <= 2.9e+38)) {
		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 <= (-115000.0d0)) .or. (.not. (z <= 2.9d+38))) 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 <= -115000.0) || !(z <= 2.9e+38)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -115000.0) or not (z <= 2.9e+38):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -115000.0) || !(z <= 2.9e+38))
		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 <= -115000.0) || ~((z <= 2.9e+38)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -115000.0], N[Not[LessEqual[z, 2.9e+38]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -115000 or 2.90000000000000007e38 < z

   1. Initial program 69.9%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in y around 0 94.0%

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

   if -115000 < z < 2.90000000000000007e38

   1. Initial program 88.3%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in x around inf 80.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -115000 \lor \neg \left(z \leq 2.9 \cdot 10^{+38}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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 / ((t * (y / z)) - (z * 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.5%

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

   1. associate-/l* 86.3%

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

   2. associate-*r* 86.3%

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

   3. associate-*l* 86.3%

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

   4. *-commutative 86.3%

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

   5. clear-num 86.0%

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

   6. un-div-inv 86.0%

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

   7. *-commutative 86.0%

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

   8. associate-*l* 85.9%

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

   9. pow2 85.9%

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

4. Applied egg-rr 85.9%

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

5. Taylor expanded in y around inf 97.1%

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

   1. +-commutative 97.1%

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

   2. mul-1-neg 97.1%

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

   3. unsub-neg 97.1%

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

7. Simplified 97.1%

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

8. Taylor expanded in y around 0 94.0%

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

   1. *-commutative 94.0%

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

   2. associate-*r/ 97.2%

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

   3. associate-*r/ 94.0%

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

   4. *-commutative 94.0%

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

   5. +-commutative 94.0%

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

   6. mul-1-neg 94.0%

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

   7. *-commutative 94.0%

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

   8. associate-*r/ 97.2%

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

   9. sub-neg 97.2%

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

   10. associate-*r/ 94.0%

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

   11. *-commutative 94.0%

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

   12. associate-*r/ 95.6%

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

10. Simplified 95.6%

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

11. Final simplification 95.6%

    \[\leadsto x + y \cdot \frac{2}{t \cdot \frac{y}{z} - z \cdot 2} \]
12. 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 79.5%

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

3. Simplified 86.3%

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

5. Taylor expanded in x around inf 78.8%

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

6. Final simplification 78.8%

   \[\leadsto x \]
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 2024053 
(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)))))