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

Percentage Accurate: 81.2% → 96.9%

Time: 11.6s

Alternatives: 4

Speedup: 1.0×

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: 81.2% 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: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-116} \lor \neg \left(z \leq 1.55 \cdot 10^{-230}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.2e-116) (not (<= z 1.55e-230)))
   (- x (/ (* y 2.0) (- (* z 2.0) (* t (/ y z)))))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.2e-116) || !(z <= 1.55e-230)) {
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (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 <= (-2.2d-116)) .or. (.not. (z <= 1.55d-230))) then
        tmp = x - ((y * 2.0d0) / ((z * 2.0d0) - (t * (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 <= -2.2e-116) || !(z <= 1.55e-230)) {
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))));
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.2e-116) or not (z <= 1.55e-230):
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))))
	else:
		tmp = x - ((z * -2.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.2e-116) || !(z <= 1.55e-230))
		tmp = Float64(x - Float64(Float64(y * 2.0) / Float64(Float64(z * 2.0) - Float64(t * 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 <= -2.2e-116) || ~((z <= 1.55e-230)))
		tmp = x - ((y * 2.0) / ((z * 2.0) - (t * (y / z))));
	else
		tmp = x - ((z * -2.0) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.2e-116], N[Not[LessEqual[z, 1.55e-230]], $MachinePrecision]], N[(x - N[(N[(y * 2.0), $MachinePrecision] / N[(N[(z * 2.0), $MachinePrecision] - N[(t * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * -2.0), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2000000000000001e-116 or 1.55e-230 < z

   1. Initial program 80.8%

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

   3. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. un-div-inv 90.2%

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

      3. *-commutative 90.2%

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

      4. *-commutative 90.2%

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

      5. associate-*l* 90.2%

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

      6. pow2 90.2%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in y around 0 96.0%

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

      1. +-commutative 96.0%

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

      2. mul-1-neg 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r/ 98.1%

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

      5. unsub-neg 98.1%

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

      6. *-commutative 98.1%

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

      7. associate-*r/ 96.0%

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

      8. *-commutative 96.0%

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

      9. associate-/l* 98.0%

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

   9. Simplified 98.0%

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

   if -2.2000000000000001e-116 < z < 1.55e-230

   1. Initial program 87.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-116} \lor \neg \left(z \leq 1.55 \cdot 10^{-230}\right):\\ \;\;\;\;x - \frac{y \cdot 2}{z \cdot 2 - t \cdot \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-267}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-230}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -5.6e-95)
     t_1
     (if (<= z 5.1e-267)
       x
       (if (<= z 1.6e-230) (* 2.0 (/ z t)) (if (<= z 6.8e+158) x t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -5.6e-95) {
		tmp = t_1;
	} else if (z <= 5.1e-267) {
		tmp = x;
	} else if (z <= 1.6e-230) {
		tmp = 2.0 * (z / t);
	} else if (z <= 6.8e+158) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x - (y / z)
    if (z <= (-5.6d-95)) then
        tmp = t_1
    else if (z <= 5.1d-267) then
        tmp = x
    else if (z <= 1.6d-230) then
        tmp = 2.0d0 * (z / t)
    else if (z <= 6.8d+158) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -5.6e-95) {
		tmp = t_1;
	} else if (z <= 5.1e-267) {
		tmp = x;
	} else if (z <= 1.6e-230) {
		tmp = 2.0 * (z / t);
	} else if (z <= 6.8e+158) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	tmp = 0
	if z <= -5.6e-95:
		tmp = t_1
	elif z <= 5.1e-267:
		tmp = x
	elif z <= 1.6e-230:
		tmp = 2.0 * (z / t)
	elif z <= 6.8e+158:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -5.6e-95)
		tmp = t_1;
	elseif (z <= 5.1e-267)
		tmp = x;
	elseif (z <= 1.6e-230)
		tmp = Float64(2.0 * Float64(z / t));
	elseif (z <= 6.8e+158)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	tmp = 0.0;
	if (z <= -5.6e-95)
		tmp = t_1;
	elseif (z <= 5.1e-267)
		tmp = x;
	elseif (z <= 1.6e-230)
		tmp = 2.0 * (z / t);
	elseif (z <= 6.8e+158)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.6e-95], t$95$1, If[LessEqual[z, 5.1e-267], x, If[LessEqual[z, 1.6e-230], N[(2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+158], x, t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5999999999999998e-95 or 6.7999999999999998e158 < z

   1. Initial program 75.8%

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

   3. Simplified 87.3%

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

   5. Taylor expanded in y around 0 88.9%

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

   if -5.5999999999999998e-95 < z < 5.10000000000000009e-267 or 1.6e-230 < z < 6.7999999999999998e158

   1. Initial program 88.6%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in x around inf 77.0%

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

   if 5.10000000000000009e-267 < z < 1.6e-230

   1. Initial program 71.2%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in y around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 78.7%

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

      1. *-commutative 78.7%

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

   10. Simplified 78.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.6 \cdot 10^{-95}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-267}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-230}:\\ \;\;\;\;2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+158}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-95} \lor \neg \left(z \leq 18000000000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6e-95) (not (<= z 18000000000000.0)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6e-95) || !(z <= 18000000000000.0)) {
		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 <= (-6d-95)) .or. (.not. (z <= 18000000000000.0d0))) 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 <= -6e-95) || !(z <= 18000000000000.0)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6e-95) or not (z <= 18000000000000.0):
		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 <= -6e-95) || !(z <= 18000000000000.0))
		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 <= -6e-95) || ~((z <= 18000000000000.0)))
		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, -6e-95], N[Not[LessEqual[z, 18000000000000.0]], $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 < -6e-95 or 1.8e13 < z

   1. Initial program 76.3%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in y around 0 85.7%

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

   if -6e-95 < z < 1.8e13

   1. Initial program 90.5%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in y around inf 93.2%

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

      1. associate-*r/ 93.2%

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

      2. *-commutative 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-95} \lor \neg \left(z \leq 18000000000000\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 82.3%

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

3. Simplified 88.6%

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

5. Taylor expanded in x around inf 72.8%

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

6. Final simplification 72.8%

   \[\leadsto x \]
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 2024076 
(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)))))