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

Percentage Accurate: 81.2% → 92.9%

Time: 6.5s

Alternatives: 4

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: 81.2% 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: 92.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot t - z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{t\_1} \leq 10^{+220}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y t) (* z (* 2.0 z)))))
   (if (<= (+ x (/ (* (* y 2.0) z) t_1)) 1e+220)
     (+ x (* (* y 2.0) (/ z t_1)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * t) - (z * (2.0 * z));
	double tmp;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1e+220) {
		tmp = x + ((y * 2.0) * (z / t_1));
	} else {
		tmp = x - (y / z);
	}
	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 = (y * t) - (z * (2.0d0 * z))
    if ((x + (((y * 2.0d0) * z) / t_1)) <= 1d+220) then
        tmp = x + ((y * 2.0d0) * (z / t_1))
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * t) - (z * (2.0 * z));
	double tmp;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1e+220) {
		tmp = x + ((y * 2.0) * (z / t_1));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * t) - (z * (2.0 * z))
	tmp = 0
	if (x + (((y * 2.0) * z) / t_1)) <= 1e+220:
		tmp = x + ((y * 2.0) * (z / t_1))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * t) - Float64(z * Float64(2.0 * z)))
	tmp = 0.0
	if (Float64(x + Float64(Float64(Float64(y * 2.0) * z) / t_1)) <= 1e+220)
		tmp = Float64(x + Float64(Float64(y * 2.0) * Float64(z / t_1)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * t) - (z * (2.0 * z));
	tmp = 0.0;
	if ((x + (((y * 2.0) * z) / t_1)) <= 1e+220)
		tmp = x + ((y * 2.0) * (z / t_1));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * t), $MachinePrecision] - N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], 1e+220], N[(x + N[(N[(y * 2.0), $MachinePrecision] * N[(z / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t)))) < 1e220

   1. Initial program 96.2%

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

   3. Simplified 96.6%

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

   if 1e220 < (-.f64 x (/.f64 (*.f64 (*.f64 y 2) z) (-.f64 (*.f64 (*.f64 z 2) z) (*.f64 y t))))

   1. Initial program 32.3%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in y around 0 82.7%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)} \leq 10^{+220}:\\ \;\;\;\;x + \left(y \cdot 2\right) \cdot \frac{z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-51} \lor \neg \left(z \leq 5.6 \cdot 10^{-61}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.25e-51) (not (<= z 5.6e-61)))
   (- x (/ y z))
   (- x (/ (* z -2.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.25e-51) || !(z <= 5.6e-61)) {
		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 <= (-1.25d-51)) .or. (.not. (z <= 5.6d-61))) 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 <= -1.25e-51) || !(z <= 5.6e-61)) {
		tmp = x - (y / z);
	} else {
		tmp = x - ((z * -2.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.25e-51) or not (z <= 5.6e-61):
		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 <= -1.25e-51) || !(z <= 5.6e-61))
		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 <= -1.25e-51) || ~((z <= 5.6e-61)))
		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, -1.25e-51], N[Not[LessEqual[z, 5.6e-61]], $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 < -1.25000000000000001e-51 or 5.6000000000000002e-61 < z

   1. Initial program 77.6%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 85.8%

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

   if -1.25000000000000001e-51 < z < 5.6000000000000002e-61

   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 90.0%

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

   5. Taylor expanded in y around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. associate-*l/ 93.5%

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

   7. Simplified 93.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-51} \lor \neg \left(z \leq 5.6 \cdot 10^{-61}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot -2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-47} \lor \neg \left(z \leq 5.6 \cdot 10^{-61}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.9e-47) (not (<= z 5.6e-61))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.9e-47) || !(z <= 5.6e-61)) {
		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 <= (-1.9d-47)) .or. (.not. (z <= 5.6d-61))) 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 <= -1.9e-47) || !(z <= 5.6e-61)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.9e-47) or not (z <= 5.6e-61):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.9e-47) || !(z <= 5.6e-61))
		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 <= -1.9e-47) || ~((z <= 5.6e-61)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.9e-47], N[Not[LessEqual[z, 5.6e-61]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -1.90000000000000007e-47 or 5.6000000000000002e-61 < z

   1. Initial program 77.3%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in y around 0 85.5%

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

   if -1.90000000000000007e-47 < z < 5.6000000000000002e-61

   1. Initial program 89.3%

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

   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. Taylor expanded in x around inf 84.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-47} \lor \neg \left(z \leq 5.6 \cdot 10^{-61}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% 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 82.9%

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

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 77.2%

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

6. Final simplification 77.2%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 99.8% 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 2024040 
(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)))))